Volume 14, Issue 1

In this article, we consider the incompressible limit of global-in-time strong solutions with arbitrary large initial velocity for the compressible viscoelastic fluids in the sense of critical Besov framework. We decouple our compressible system into two coupling sub-systems by introducing a skew symmetric matrix, which is related to the deformation tensor. This work generalizes the similar result obtained by Hu et al. ( Incompressible limit for compressible viscoelastic flows with large velocity , Advances in Nonlinear Analysis 12 (2023), 20220324) to the critical functional space with respective to the natural scaling of the system. The proof relies on the dispersive property of the linear system on the high-frequency regime and the parabolic property on the low-frequency regime. The dispersion tends to disappear when λ \lambda tends to infinite, but having large λ \lambda provides strong dissipation on the potential part of the velocity and thus makes the flow almost incompressible. In addition, by exploiting the intrinsic structure of the viscoelastic system, we obtain the global uniform estimates of the solutions near equilibrium.

In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p -Laplacian in R 3 {{\mathbb{R}}}^{3} of the form: ε s p ( − Δ ) p s u + V ( x ) ∣ u ∣ p − 2 u − ϕ ∣ u ∣ p s ♯ − 2 u = ∣ u ∣ p s * − 2 u + f ( u ) in R 3 , ε s p ( − Δ ) s ϕ = ∣ u ∣ p s ♯ in R 3 , \left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{\mathcal{V}}\left(x){| u| }^{p-2}u-\phi {| u| }^{{p}_{s}^{\sharp }-2}u={| u| }^{{p}_{s}^{* }-2}u+f\left(u)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{sp}{\left(-\Delta )}^{s}\phi ={| u| }^{{p}_{s}^{\sharp }}\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a positive parameter, s ∈ ( 3 4 , 1 ) s\in \left(\frac{3}{4},1) , ( − Δ ) p s {\left(-\Delta )}_{p}^{s} is the fractional p -Laplacian operator, p ∈ ( 3 2 , 3 ) p\in \left(\frac{3}{2},3) , p s * = 3 p 3 − s p {p}_{s}^{* }=\frac{3p}{3-sp} is the Sobolev critical exponent, p s ♯ = p 2 ( 3 + 2 s ) ( 3 − s p ) {p}_{s}^{\sharp }=\frac{p}{2}\frac{\left(3+2s)}{\left(3-sp)} is the upper exponent in the sense of the Hardy-Littlewood-Sobolev inequality, V ( x ) : R 3 → R {\mathcal{V}}\left(x):{{\mathbb{R}}}^{3}\to {\mathbb{R}} symbolizes a continuous potential function with a local minimum, and the continuous function f f possesses subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann category theory, we use the topological arguments to attain the multiplicity and concentration of the positive solutions for the above system.

In this article, we study the behavior of multiple continua of solutions to the semilinear elliptic problem − Δ u = λ f ( u ) , in Ω , u = 0 , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=\lambda f\left(u),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω \Omega is a bounded open subset of R N {{\mathbb{R}}}^{N} and f f is a nonnegative continuous real function with multiple zeros. We analyze both the behavior of unbounded continua of solutions having norm between consecutive zeros of f f , and the asymptotic behavior of the multiple unbounded continua in the case in which f f has a countable infinite set of positive zeros. In both cases, we pay special attention to the multiplicity results they give rise to. For the model cases f ( t ) = t r ( 1 + sin t ) f\left(t)={t}^{r}\left(1+\sin t) and f ( t ) = t r 1 + sin 1 t f\left(t)={t}^{r}\left(\phantom{\rule[-0.75em]{}{0ex}},1+\sin \frac{1}{t}\right) with r ≥ 0 r\ge 0 , we show the surprising fact that there are some values of r r for which every λ > 0 \lambda \gt 0 is a bifurcation point (either from infinity or from zero) that is not a branching point.

In this article, we study the following nonlinear Dirac equation: − i α ⋅ ∇ u + a β u + V ( x ) u = g ( x , ∣ u ∣ ) u , x ∈ R 3 . -i\alpha \hspace{0.33em}\cdot \hspace{0.33em}\nabla u+a\beta u+V\left(x)u=g\left(x,| u| )u,\hspace{1em}x\in {{\mathbb{R}}}^{3}. The Dirac operator is unbounded from below and above so the associated energy functional is strongly indefinite. Compared with some existing issues, the most interesting problem in this article is that we assume that the nonlinearity satisfies a local super-quadratic growth condition, which is weaker than the usual global version. This case allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. Using variational tools together with deformation lemma, genus theory, and some special techniques, we prove the existence of ground-state solutions and infinitely many geometrically distinct solutions under local super-quadratic growth condition. Our results complement and extend the related results of Bartsch and Ding [ Solutions of nonlinear Dirac equations , J. Differ. Equ. 226 (2006), 210–249] and of Chen et al. [ Ground state solutions for the Dirac equation with periodic external force field , Sci. Sin. Math. (Chinese) 54 (2024), 747–764].

We prove the existence of entire, radial, and signed bounded solutions for a quasilinear elliptic equation in R N {{\mathbb{R}}}^{N} driven by a Leray-Lions operator of the ( p, q )-type. For this, we need an extension of related results by Boccardo-Murat-Puel and a variational approach in intersections of Banach spaces introduced by Candela-Palmieri.

The concept of combinatorial p th Calabi flow is proposed for finding circle packing metrics with prescribed curvatures. It has been studied under the Euclidean and hyperbolic background geometries. This study investigates the combinatorial p th Calabi flow for total geodesic curvatures under the spherical background geometry. It is established that the combinatorial p th Calabi flow, subject to given total geodesic curvatures under the spherical background geometry, exists for all time t ∈ [ 0 , + ∞ ) t\in \left[0,+\infty ) and converges to an ideal circle pattern metric. This finding introduces a new algorithm for constructing ideal circle pattern metrics with prescribed total geodesic curvatures.

We derive various sharp upper bounds for the p p -capacity of a smooth compact set K K in the hyperbolic space H n {{\mathbb{H}}}^{n} and the Euclidean space R n {{\mathbb{R}}}^{n} . First, by using the inverse mean curvature flow, for the mean convex and star-shaped set K K in H n {{\mathbb{H}}}^{n} , we obtain sharp upper bounds for the p p -capacity Cap p ( K ) {{\rm{Cap}}}_{p}\left(K) in three cases: (1) n ≥ 2 n\ge 2 and p = 2 p=2 , (2) n = 2 n=2 and p ≥ 3 p\ge 3 , and (3) n = 3 n=3 and 1 < p ≤ 3 1\lt p\le 3 ; using the unit-speed normal flow, we prove a sharp upper bound for Cap p ( K ) {{\rm{Cap}}}_{p}\left(K) of a convex set K K in H n {{\mathbb{H}}}^{n} for n ≥ 2 n\ge 2 and p > 1 p\gt 1 . Second, for the compact set K K in R 3 {{\mathbb{R}}}^{3} , using the weak inverse mean curvature flow, we obtain a sharp upper bound for the p p -capacity ( 1 < p < 3 1\lt p\lt 3 ) of the set K K with connected boundary; by using the inverse anisotropic mean curvature flow, we deduce a sharp upper bound for the anisotropic p p -capacity ( 1 < p < 3 1\lt p\lt 3 ) of an F F -mean convex and star-shaped set K K in R 3 {{\mathbb{R}}}^{3} .

In this article, we study the following stationary Schrödinger equation on hyperbolic space: − Δ H N u + λ u = f ( u ) , x ∈ H N , N ≥ 3 , -{\Delta }_{{{\mathbb{H}}}^{N}}u+\lambda u=f\left(u),\hspace{1.0em}x\in {{\mathbb{H}}}^{N},\hspace{1em}N\ge 3, where Δ H N {\Delta }_{{{\mathbb{H}}}^{N}} denotes the Laplace-Beltrami operator on H N {{\mathbb{H}}}^{N} , λ ∈ R \lambda \in {\mathbb{R}} , and f f is locally Lipschitz continuous satisfying asymptotically linear at infinity. After a reduction from hyperbolic case to Euclidean case, using variational methods, we prove the existence of positive solutions for the aforementioned equation under suitable conditions on λ \lambda and f f .

The present study investigates the existence and non-degeneracy of normalized solutions for the following fractional Schrödinger equation: ( − Δ ) s u + V ( x ) u = a u p + μ u , x ∈ R N , u ∈ H s ( R N ) {\left(-\Delta )}^{s}u+V\left(x)u=a{u}^{p}+\mu u,\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}u\in {H}^{s}\left({{\mathbb{R}}}^{N}) with the L 2 {L}^{2} -restriction ∫ R N u 2 ( x ) d x = 1 {\int }_{{{\mathbb{R}}}^{N}}{u}^{2}\left(x){\rm{d}}x=1 , where s ∈ ( 0 , 1 ) s\in \left(0,1) , p ∈ ( 1 , 2 N N − 2 s − 1 ) p\in \left(1,\frac{2N}{N-2s}-1) , N > 2 s N\gt 2s , a > 0 a\gt 0 and V ( x ) V\left(x) is some smooth trapping potential. Via a Lyapunov-Schmidt variational reduction, we first construct solutions of the form u a ∼ − μ a a 1 p − 1 ∑ j = 1 k U ( − μ a ) 1 2 s ( x − x a , j ) , {u}_{a} \sim {\left(\frac{-{\mu }_{a}}{a}\right)}^{\tfrac{1}{p-1}}\mathop{\sum }\limits_{j=1}^{k}U\left(\phantom{\rule[-0.75em]{}{0ex}},{\left(-{\mu }_{a})}^{\tfrac{1}{2s}}\left(x-{x}_{a,j})\right), where x a , j {x}_{a,j} approach suitable critical points of V ( x ) V\left(x) , U ( x ) ∈ H s ( R N ) U\left(x)\in {H}^{s}\left({{\mathbb{R}}}^{N}) is the unique radially symmetric positive ground state solution of ( − Δ ) s u + u = u p , u ( 0 ) = max x ∈ R N u ( x ) {\left(-\Delta )}^{s}u+u={u}^{p},u\left(0)={\max }_{x\in {{\mathbb{R}}}^{N}}u\left(x) . Subsequently, the local Pohozaev identity techniques are applied to establish the non-degeneracy of such normalized solutions. This study successfully addresses the complexities arising from the non-local characteristics of the fractional Laplacian in the local analysis and pointwise estimates of solutions. In contrast to the unconstrained scenario, the mass-critical power, denoted as p = 4 s N + 1 p=\frac{4s}{N}+1 , acts as a pivotal threshold. It delineates distinct ranges of values for p p , each corresponding to vastly different concentration behaviors exhibited by the solutions. This phenomenon unequivocally underscores the profound impact of constraint conditions on the intricate dynamics of the solutions.

In this article, we investigate non-coercive variational equations under assumptions related to generalized monotonicity. We present some general abstract tools regarding the existence of bounded solutions and their multiplicity, which we then apply to the Dirichlet boundary value problem driven by the perturbed (negative) p p -Laplacian. As a by-product of our findings, we provide a version of the Browder-Minty theorem in the potential case that does not involve the Brouwer fixed-point theorem but utilizes optimization techniques.

This article focuses on the study of the following Schrödinger-Poisson system with zero mass: − Δ u + ϕ u = ∣ u ∣ u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+\phi u=| u| u+f\left(u),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. where f f is a continuous function satisfying some general growth conditions, and it requires only to be super-quadratic growth at infinity and includes, in particular, the pure power function ∣ u ∣ p − 2 u {| u| }^{p-2}u with p ∈ ( 3 , 6 ) p\in \left(3,6) . The nonlinear term ∣ u ∣ u | u| u is the so-called Coulomb critical nonlinearity because it presents a certain scaling invariance and the mountain-pass geometry cannot be established when f = 0 f=0 . Few results are known to such case. By developing some delicate analyses and using detailed estimates, we obtain the existence of ground-states and least energy solution for the aforementioned system under some natural assumptions on f f .

In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) \phi :\left[0,\infty )\to \left[0,\infty ) . Given any bounded domain with the slice property, the authors prove that it is a generalized Hajłasz-Sobolev embedding domain if and only if it is a generalized F F -weak cigar domain, where F F is a modulus of continuity related to the weight function ϕ \phi of the generalized Hajłasz-Sobolev spaces under consideration. Comparing with the classical Hajłasz-Soblev spaces, one main difficulty in dealing with generalized Hajłasz-Sobolev spaces lies in that both its smoothness weight function ϕ \phi and the related modulus of continuity F F have no explicit expression. To overcome this, the authors introduce and use some key indices to accurately describe the increasing or the decreasing behavior of both ϕ \phi and F F . Besides the classical Hajłasz-Sobolev spaces, this result can be applied to many other nontrivial spaces such as Hajłasz-Sobolev spaces with logarithmic smoothness and is of wide generality.

The aim of this article is to study subharmonic solutions of superquadratic and asymptotically (constant) linear nonautonomous Hamiltonian systems in R 2 n {{\mathbb{R}}}^{2n} respectively, and to improve the results in Professor Liu’s [ Subharmonic solutions of Hamiltonian systems , Nonlinear Anal. 42 (2000), no. 2, 185–198, Ser. A: Theory Methods]. For the T T -periodic Hamiltonian system, we show that for each integer j ≥ 1 j\ge 1 , there exists a nonconstant j T jT -periodic solution x j {x}_{j} such that x j {x}_{j} and x p j {x}_{pj} ( p ∈ N p\in {\mathbb{N}} ) are geometrically distinct provided p > 2 n p\gt 2n ; for n = 1 n=1 , x j {x}_{j} and x p j {x}_{pj} ( p ∈ N p\in {\mathbb{N}} ) are geometrically distinct provided p > 1 p\gt 1 ; if x j {x}_{j} is nondegenerate, then x j {x}_{j} and x p j {x}_{pj} ( p ∈ N p\in {\mathbb{N}} ) are geometrically distinct provided p > 1 p\gt 1 .

It is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done. In this study, we establish the ( m + 1 ) \left(m+1) -expansions of entire k k -convex large solutions near infinity to the k k -Hessian equation S k ( D 2 u ) = b ( x ) f ( u ) in R N , {S}_{k}\left({D}^{2}u)=b\left(x)f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where m ∈ N + m\in {{\mathbb{N}}}_{+} , 1 ≤ k < N ⁄ 2 1\le k\lt N/2 , N ≥ 3 N\ge 3 , f ( u ) = u p f\left(u)={u}^{p} ( p > k p\gt k ) near infinity or f ( u ) = u p + u q f\left(u)={u}^{p}+{u}^{q} ( p > k p\gt k and p > q > − 1 p\gt q\gt -1 ) near infinity. In particular, inspired by some ideas in partition theory of integer, we give a recursive formula of the coefficient of ( n + 1 ) \left(n+1) -order terms ( 2 ≤ n ≤ m ) \left(2\le n\le m) of the expansions. And if f ( u ) = u p + u q f\left(u)={u}^{p}+{u}^{q} near infinity, we reveal the influence of the lower term of f ( u ) f\left(u) on the expansion of any entire large solution.

This article is concerned with the stability of time-periodic traveling fronts for reaction–diffusion equations with time-periodic degenerate monostable and ignition nonlinearities. Based on the existence of pyramidal traveling fronts established by Bu et al., we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces and establish some important estimates. Then, we show the asymptotic stability of pyramidal traveling fronts by using the sub-super solution method combined with the comparison principle.

We study the following fractional Kirchhoff-Choquard equation: a + b ∫ R N ( − Δ ) s 2 u 2 d x ( − Δ ) s u + V ( x ) u = ( I μ * F ( u ) ) f ( u ) , x ∈ R N , u ∈ H s ( R N ) , \left\{\begin{array}{l}\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\left|{\left(-\Delta )}^{\frac{s}{2}}u\right|}^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u+V\left(x)u=\left({I}_{\mu }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where a , b a,b are positive constants, N > 2 s N\gt 2s , μ ∈ ( ( N − 4 s ) + , N ) \mu \in \left({\left(N-4s)}_{+},N) , s ∈ ( 0 , 1 ) s\in \left(0,1) , and I μ {I}_{\mu } is the Riesz potential. Considering the case that nonlinearity f f has critical growth, combining a monotonicity trick and global compactness lemma, we prove that the equation has a ground-state solution. Moreover, we study the regularity of ground-state solutions to the above equation, which extends the results in Moroz-Van Schaftingen [ Existence of groundstates for a class of nonlinear Choquard equations , Trans. Amer. Math. Soc. 367 (2015), 6557–6579] to the fractional Laplacian case.

In this study, we deal with a multivalued elliptic variational inequality involving a logarithmic perturbed variable exponents double-phase operator. Additionally, it features a multivalued convection term alongside two multivalued terms, one defined within the domain and the other on its boundary. Under the noncoercive framework, we establish the existence results of weak solutions for the multivalued inequality by employing a surjective theorem for multivalued pseudomonotone operators along with the penalty technique. On the other hand, we prove the compactness of solution set by employing the S + {S}_{+} -property of the associated perturbed variable exponent double-phase operator. Finally, we focus on special cases to the multivalued inequality, where K K is a bilateral constraint set, and the two multivalued terms are Clarke’s generalized gradients with respect to two locally Lipschitz functions.

By using critical point theory in combination with periodic approximations, we obtain novel sufficient conditions for the existence of nontrivial homoclinic solutions for a class of periodic partial difference equations with sign-changing mixed nonlinearities. Notably, in certain specific cases, a necessary and sufficient condition is also established.

We consider the compressible Navier-Stokes-Korteweg equations describing the dynamics of a liquid-vapor mixture with diffuse interphase in R d {{\mathbb{R}}}^{d} with d ≥ 3 d\ge 3 when the initial perturbation is suitably small. In particular, when the base sound speed P ′ ( ρ ¯ ) = 0 \sqrt{P^{\prime} \left(\bar{\rho })}=0 , we first give the global existence and optimal L 2 {L}^{2} -decay rate of the smooth solution, where the optimality means that the decay rate of the solution is the same as that for the corresponding linearized system, and there is no decay loss for the highest-order spatial derivatives of the solution. Then, we establish space-time behavior of the solution based on Green’s function method. It is obviously different from the case P ′ ( ρ ¯ ) > 0 \sqrt{P^{\prime} \left(\bar{\rho })}\gt 0 , which obeys the generalized Huygens’ principle as the compressible Navier-Stokes equations.

We consider the properties of minimizers for the following constraint minimization problem: i ( c ) ≔ inf u ∈ S 1 I c ( u ) , i\left(c):= \mathop{\inf }\limits_{u\in {S}_{1}}{I}_{c}\left(u), where the L 2 {L}^{2} -unite sphere S 1 = { u ∈ H 1 ( R N ) ∣ ∫ R N V ( x ) u 2 d x < + ∞ , ‖ u ‖ L 2 = 1 } , {S}_{1}=\left\{u\in {H}^{1}\left({{\mathbb{R}}}^{N})| {\int }_{{{\mathbb{R}}}^{N}}V\left(x){u}^{2}{\rm{d}}x\lt +\infty ,{\Vert u\Vert }_{{L}^{2}}=1\right\}, and the energy functional I c ( u ) {I}_{c}\left(u) is defined by I c ( u ) = a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 + 1 2 ∫ R N V ( x ) u 2 d x − c p + 2 ∫ R N ∣ u ∣ p + 2 d x . {I}_{c}\left(u)=\frac{a}{2}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{| \nabla u| }^{2}{\rm{d}}x+\frac{b}{4}{\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{| \nabla u| }^{2}{\rm{d}}x\right)}^{2}+\frac{1}{2}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x){u}^{2}{\rm{d}}x-\frac{c}{p+2}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{p+2}{\rm{d}}x. If V ( x ) ≡ 0 V\left(x)\equiv 0 , we shall show the uniqueness of minimizers of i ( c ) i\left(c) for c > c ˜ p p ∈ 0 , 4 N c\gt {\widetilde{c}}_{p}\hspace{0.33em}\left(\phantom{\rule[-0.75em]{}{0ex}},p\in \left(\phantom{\rule[-0.75em]{}{0ex}},0,\frac{4}{N}\right]\right) , or c ≥ c ˜ p p ∈ 4 N , 8 N c\ge {\widetilde{c}}_{p}\hspace{0.33em}\left(\phantom{\rule[-0.75em]{}{0ex}},p\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{4}{N},\frac{8}{N}\right)\right) ; if V ( x ) ≢ 0 V\left(x)\not\equiv 0 , the concentration behavior of minimizers for i ( c ) i\left(c) is discussed as c → + ∞ . c\to +\infty . Moreover, the uniqueness of minimizers for i ( c ) i\left(c) is also proved as c c large enough, which extends the related results.

The Keller-Segel-Navier-Stokes system in R N {{\mathbb{R}}}^{N} is considered, where N ≥ 3 N\ge 3 . We show the existence and uniqueness of local mild solutions for arbitrary initial data and gravitational potential in scaling invariant Lorentz spaces. Although such a result has already been shown by Kozono, Miura, and Sugiyama ( Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid , J. Funct. Anal. 270 (2016), no. 5, 1663–1683), we reveal the precise regularities of mild solutions by showing the smoothing estimates of the heat semigroup on Lorentz spaces. The method is based on the real interpolation. In addition, we prove that the mild solutions exist globally in time, provided that the initial data are sufficiently small. Compared with the usual result, a part of the smallness conditions is reduced. We also obtain the asymptotic behavior of the global mild solutions. In the proof of the asymptotic behavior, to overcome a lack of density for the space L ∞ ( R N ) {L}^{\infty }\left({{\mathbb{R}}}^{N}) to which one c 0 {c}_{0} of the initial data belongs, we show the decay of the global solutions without any approximation for c 0 {c}_{0} .

In this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i ∂ t u = − Δ u + 2 V ( x ) u + 2 Ω L z u − λ u − b u ∫ R N ∣ u ( y ) ∣ 2 ∣ x − y ∣ 2 d y , ( t , x ) ∈ R × R N , u ( 0 , x ) = u 0 ( x ) , \left\{\begin{array}{l}i{\partial }_{t}u=-\Delta u+2V\left(x)u+2\Omega {L}_{z}u-\lambda u-bu\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u(y)| }^{2}}{{| x-y| }^{2}}{\rm{d}}y,\hspace{1em}\left(t,x)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\left(0,x)={u}_{0}\left(x),\hspace{1.0em}\end{array}\right. where N ≥ 3 N\ge 3 , b > 0 b\gt 0 , V ( x ) = ∣ x ∣ 2 2 V\left(x)=\frac{{| x| }^{2}}{2} , L z {L}_{z} is the angular momentum operator with the critical rotational speed Ω = 1 \Omega =1 , and the constant λ \lambda is the unknown Lagrange multiplier. We prove that the L 2 {L}^{2} -constraint minimizers exist if and only if the parameter b b satisfies b < b * = ‖ U ‖ 2 2 b\lt {b}_{* }={\Vert U\Vert }_{2}^{2} , where U U is a positive radially symmetric ground state of − Δ u + u − u ∫ R N u 2 ( y ) ∣ x − y ∣ 2 d y = 0 -\Delta u+u-u{\int }_{{{\mathbb{R}}}^{N}}\frac{{u}^{2}(y)}{{| x-y| }^{2}}{\rm{d}}y=0 in R N {{\mathbb{R}}}^{N} . We also establish the orbital stability result of prescribed mass standing waves for the equation when b < b * b\lt {b}_{* } . When b b approaches b * {b}_{* } , the system collapses to a profile obtained from the optimizer of a Gagliardo-Nirenberg inequality.

In this article, we mainly study the qualitative properties of solutions for dual fractional-order parabolic equations with nonlocal Monge-Ampère operators in different domains ∂ t β μ ( y , t ) − D α τ μ ( y , t ) = f ( μ ( y , t ) ) . {\partial }_{t}^{\beta }\mu \left(y,t)-{D}_{\alpha }^{\tau }\mu \left(y,t)=f\left(\mu \left(y,t)). We first establish a series of maximum principles and averaging effects theorems for antisymmetric functions and then used the method of moving planes and sliding planes to establish radial symmetry, monotonicity, nonexistence, and Liouville theorem for positive solutions.

We deal with non-uniformly elliptic integral functionals w ↦ ∫ c ( x ) ∣ D w ∣ log ( 1 + ∣ D w ∣ ) + a ( x ) ( ∣ D w ∣ 2 + s 2 ) q 2 + 1 d x , w\mapsto \int \left[{\mathfrak{c}}\left(x)| Dw| \log \left(1+| Dw| )+a\left(x){\left({| Dw| }^{2}+{s}^{2})}^{\tfrac{q}{2}}+1\right]{\rm{d}}x, with s ∈ [ 0 , 1 ] s\in \left[0,1] , q > 1 q\gt 1 , 0 ≤ c ( ⋅ ) ≤ Λ , a ( ⋅ ) ≥ 0 0\le {\mathfrak{c}}\left(\cdot )\le \Lambda ,a\left(\cdot )\ge 0 , and a ( x ) + c ( x ) ≥ 1 Λ a\left(x)+{\mathfrak{c}}\left(x)\ge \frac{1}{\Lambda } for some Λ > 0 \Lambda \gt 0 . In this article, we establish that the gradient of a local minimizer for the aforementioned functionals is locally Hölder continuous.

Let G {\mathcal{G}} be a stratified Lie group, and let { X j } 1 ≤ j ≤ n 1 {\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector fields of degree one on G {\mathcal{G}} and Δ = − ∑ j = 1 n 1 X j 2 \Delta =-{\sum }_{j=1}^{{n}_{1}}{X}_{j}^{2} be the sub-Laplacian of G {\mathcal{G}} . Given that 0 ≤ α < Q 0\le \alpha \lt {\mathbb{Q}} , this article studies the commutators [ b , X j Δ − 1 + α 2 ] m , j = 1 , … , n 1 {\left[b,{X}_{j}{\Delta }^{-\frac{1+\alpha }{2}}]}^{m},\hspace{0.33em}j=1,\ldots ,{n}_{1} of order m ∈ N m\in {\mathbb{N}} and establishes their uniform two-weight boundedness from L p ( μ p ) {L}^{p}\left({\mu }^{p}) to L q ( w q ) {L}^{q}\left({w}^{q}) for any 0 < α < Q 0\lt \alpha \lt {\mathbb{Q}} and 1 q = 1 p − α Q \frac{1}{q}=\frac{1}{p}-\frac{\alpha }{{\mathbb{Q}}} via BMO ν 1 ⁄ m ( G ) {{\rm{BMO}}}_{{\nu }^{1/m}}\left({\mathcal{G}}) space, assuming that μ , w ∈ A p , q \mu ,w\in {A}_{p,q} and ν = μ w \nu =\frac{\mu }{w} . Based on this, we also give the characterization of VMO ( G ) {\rm{VMO}}\left({\mathcal{G}}) space with respect to the uniform weighted compactness of [ b , X j Δ − 1 + α 2 ] \left[b,{X}_{j}{\Delta }^{-\tfrac{1+\alpha }{2}}] for 0 ≤ α < Q 0\le \alpha \lt {\mathbb{Q}} . As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on G {\mathcal{G}} can be recovered as α → 0 \alpha \to 0 .

We consider the following elliptic problem with mixed local and nonlocal operators: ( − Δ ) s 1 u + ( − Δ ) s 2 u + λ u = f ( u ) in R N , ∫ R N ∣ u ( x ) ∣ 2 d x = c , u ∈ H s 2 ( R N ) , \left\{\begin{array}{l}{\left(-\Delta )}^{{s}_{1}}u+{\left(-\Delta )}^{{s}_{2}}u+\lambda u=f\left(u)\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u\left(x)| }^{2}{\rm{d}}x=c,\hspace{1.0em}\\ u\in {H}^{{s}_{2}}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where 0 < s 1 < s 2 < 1 0\lt {s}_{1}\lt {s}_{2}\lt 1 , ( − Δ ) s i {\left(-\Delta )}^{{s}_{i}} is fractional Laplacian. We get a sharp description of the existence and nonexistence of the global minimizer on the mass constraint, which is called energy ground state. By using of concentration-compactness principle, we show that there exists a constant c 0 {c}_{0} such that there exists an energy ground state if c > c 0 c\gt {c}_{0} and there exists no energy ground state if 0 < c < c 0 0\lt c\lt {c}_{0} . Some almost critical conditions which determine c 0 = 0 {c}_{0}=0 or c 0 > 0 {c}_{0}\gt 0 are established. In the case c = c 0 c={c}_{0} , under certain conditions, we establish the existence result of energy ground state. Finally, by using the method of mountain pass type characterization, we show that any energy ground state is action ground state of the corresponding action functional.

Let Ω ⊂ R n ( n ≥ 2 ) \Omega \subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) be either an open ball B R {B}_{R} centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form Δ p u = f 1 ( ∣ x ∣ ) g 1 ( v ) ∣ ∇ u ∣ α in Ω , Δ p v = f 2 ( ∣ x ∣ ) g 2 ( v ) h ( ∣ ∇ u ∣ ) in Ω , \left\{\begin{array}{rcl}{\Delta }_{p}u& =& {f}_{1}\left(| x| ){g}_{1}\left(v){| \nabla u| }^{\alpha }\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\Delta }_{p}v& =& {f}_{2}\left(| x| ){g}_{2}\left(v)h\left(| \nabla u| )\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. where α ≥ 0 \alpha \ge 0 , Δ p {\Delta }_{p} is the p p -Laplace operator, p > 1 p\gt 1 , and for i , j = 1 i,j=1 , 2, we assume f i , g j {f}_{i},{g}_{j} , and h h are continuous, non-negative and non-decreasing functions. For functions g j {g}_{j} which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at ∂ B R \partial {B}_{R} , and for the existence of global solutions.

This article verifies the low Mach number and non-resistive limit of local strong solutions to non-isentropic compressible magnetohydrodynamic (MHD) equations in general three-dimensional bounded domains when the temperature variation is large but finite. The uniform estimates of strong solutions are obtained in a short time interval independent of the Mach number and the magnetic resistivity coefficient, provided that the initial data are well prepared. Previous results on this multi-scale singular limit of compressible MHD equations are either for the cases of the weak solutions or for the cases of small temperature variations or for the cases of flat boundary where the mechanisms are essentially different. The new ingredient of this article is that we develop a new approach of weighted energy to establish the uniform estimates for high-order spatial derivatives of solutions.

This article is concerned with a relaxation model with degeneracy. Different from the existing studies on the regular viscous shock waves, we recognize that when the sub-characteristic condition is degenerate, the dynamical system possesses sharp viscous shock waves, a new type of viscous shock waves with sharp corners and being regionally degenerate in a half space. The regularities of these sharp viscous shock waves are further derived. Some of them can be smooth due to different degree of degeneracy, but most of them are Hölder continuous with sharp corners. Finally, the exact sharp viscous shock waves are constructed for three typical equations, and the corresponding numerical simulations with given initial data are also presented, which perfectly match the exact sharp viscous shock waves and also numerically demonstrate the stability of sharp shock waves.

In this article, we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator Δ γ p {\Delta }_{\gamma }^{p} . We extend to a generic p > 1 p\gt 1 a result, which was proved only when p = 2 p=2 . When p ≠ 2 p\ne 2 , the nonlinear operator − Δ γ p -{\Delta }_{\gamma }^{p} has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem, which is not based on linear subspaces. We use an abstract result based on a pseudo-index related to the Z 2 {{\mathbb{Z}}}_{2} -cohomological index that is applicable here. We provide a version of the Lions’ concentration-compactness principle for our operator.

In this article, we consider the influence of seasonal succession and impulsive harvesting on the dynamical behavior of solutions to a free boundary model. First, the generalized principal eigenvalue is defined and its properties are studied. Next, some sufficient conditions for spreading and for vanishing are given. Then, by introducing a one-parameter family of initial data σ ϕ \sigma \phi with σ ≥ 0 \sigma \ge 0 and ϕ \phi being a compactly supported function, we obtain a threshold value σ * {\sigma }^{* } such that spreading happens when σ > σ * \sigma \gt {\sigma }^{* } , vanishing happens when σ ≤ σ * \sigma \le {\sigma }^{* } . Finally, we prove the existence and uniqueness of T T -periodic traveling semi-wave. Moreover, when spreading persists, we use the proved T T -periodic traveling semi-wave to estimate the asymptotic speed of the free boundaries.

Let ( ℳ , g ) \left({\mathcal{ {\mathcal M} }},g) and ( K , κ ) \left({\mathcal{K}},\kappa ) be two Riemannian manifolds of dimensions N N and m m , respectively. Let ω ∈ C 2 ( ℳ ) \omega \in {C}^{2}\left({\mathcal{ {\mathcal M} }}) satisfy ω > 0 \omega \gt 0 . The warped product ℳ × ω K {\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} is the ( N + m ) \left(N+m) -dimensional product manifold ℳ × K {\mathcal{ {\mathcal M} }}\times {\mathcal{K}} equipped with the metric g + ω 2 κ g+{\omega }^{2}\kappa . We consider the following elliptic system: (1) − Δ g + ω 2 κ u + h ( x ) u = v p − α ε , in  ( ℳ × ω K , g + ω 2 κ ) , − Δ g + ω 2 κ v + h ( x ) v = u q − β ε , in  ( ℳ × ω K , g + ω 2 κ ) , u , v > 0 , in  ( ℳ × ω K , g + ω 2 κ ) , \left\{\begin{array}{ll}-\hspace{-0.03em}{\Delta }_{g+{\omega }^{2}\kappa }u+h\left(x)u={v}^{p-\alpha \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ -{\Delta }_{g+{\omega }^{2}\kappa }v+h\left(x)v={u}^{q-\beta \varepsilon },& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\\ u,v\gt 0,& \hspace{0.1em}\text{in\hspace{0.5em}}\hspace{0.1em}\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}},g+{\omega }^{2}\kappa )\text{},\end{array}\right. where Δ g + ω 2 κ = div g + ω 2 κ ∇ {\Delta }_{g+{\omega }^{2}\kappa }={{\rm{div}}}_{g+{\omega }^{2}\kappa }\nabla denotes the Laplace-Beltrami operator on ℳ × ω K {\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} , h ( x ) h\left(x) is a C 1 {C}^{1} -function on ℳ × ω K {\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} , α , β > 0 \alpha ,\beta \gt 0 are the real numbers, ε > 0 \varepsilon \gt 0 is a small parameter, and ( p , q ) ∈ ( 1 , + ∞ ) × ( 1 , + ∞ ) \left(p,q)\in \left(1,+\infty )\times \left(1,+\infty ) satisfies 1 p + 1 + 1 q + 1 = N − 2 N \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N} . For any integer k ≥ 2 k\ge 2 , using the Lyapunov-Schmidt reduction, we prove that problem (1) admits a k k -peak solution concentrated along an m m -dimensional minimal submanifold of ( ℳ × ω K ) k {\left({\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}})}^{k} as ε → 0 \varepsilon \to 0 .

This article shows time-asymptotic nonlinear stability of rarefaction wave to the Cauchy problem for the one-dimensional relaxed compressible Navier-Stokes equations with density-dependent viscosity. We prove that the solution to this typical system tends time-asymptotically to the rarefaction wave. For this, we technically construct the correction function S ˆ ( x , t ) \hat{S}\left(x,t) , which means that S ± {S}_{\pm } can be non-zero. The proof is accomplished by virtue of energy estimates.

In this article, we investigate the formation of singularities in solutions of homogeneous and inhomogeneous systems for a macroscopic production model with Chaplygin gas using the method of characteristic decomposition. It is a model originating from production planning in supply chains. Since the model with the Chaplygin gas equation of state is a strictly hyperbolic system with both linearly degenerate characteristic field and genuinely nonlinear characteristic field, we can simultaneously study the blow-up of the solution itself and the blow-up of the derivative of the solution, in which the blow-up of the solution itself corresponds to the bottlenecks in the supply chains. Besides, we obtain the C 1 {C}^{1} estimates of the solution before the solution itself blows up. In addition, we give some representative examples of numerical simulations concerning the generation of singularities of solutions.

In this article, the Cauchy problem of a compressible Navier-Stokes system with density-dependent viscosities when the initial data are spherically symmetric is considered. Firstly, we construct the classical solution for the system in B a ( t ) = { r : 0 ≤ r ≤ a ( t ) } {B}_{a\left(t)}=\left\{r:0\le r\le a\left(t)\right\} with the stress-free boundary, where a ( t ) a\left(t) is the “particle path,” which arising from a ( 0 ) = a 0 > 0 a\left(0)={a}_{0}\gt 0 . Next, the strong solution for the system with the stress-free condition on the boundary a ( t ) a\left(t) in the exterior domain R 3 \ B a ( t ) {{\mathbb{R}}}^{3}\backslash {B}_{a\left(t)} is proved by the entropy estimate and the subtle analysis on the upper and lower bounds of the density. Finally, by continuously splicing the two parts of the solution via the stress-free boundary, we establish the global strong solution of the Cauchy problem. In particular, our analysis gives a positive example that it does not exhibit vacuum states provided that no vacuum states are present initially for multi-dimensional compressible flow with density-dependent viscosities, which is consistent with Hoff and Smoller [Hoff and Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys.   216 (2001), no. 2, 255–276], where the viscosity coefficients are assumed to be positive constants for one-dimensional case.

In the article, we first investigate a population-toxicant model with nonlocal dispersal. Compared to the local dispersal (random dispersal), nonlocal dispersal brings some difficulties. One of the difficulties is that principal eigenvalue might not exist. Another difficulty is that the solution orbits lose their compactness. Motivated by previous studies, we overcome these difficulties and establish the global dynamics by spectral analysis, the methods of lower and upper solutions, and the theory of abstract monotone dynamical system. Then, we generalize these results to more general case that the dispersal strategies are mixture of local and nonlocal dispersals. Finally, we point out that we have successfully expanded the upper bound of the critical indicator given in the study by Zhou and Huang without advection terms.

We consider the α \alpha -mean curvature flow for convex graphs in Euclidean space. Given a smooth, complete, strictly convex, non-compact initial hypersurface over a strictly convex projected domain, we derive uniform curvature bounds, which are independent of the height of a graph, to give C 2 {C}^{2} -estimates for convex graphs. Consequently, these height-independent estimates imply that all the derivatives for level sets converge uniformly. Furthermore, with these estimates on level sets, the boundary of the domain of a graph, which demonstrates the behavior of level sets as the height tends to infinity, is shown to be a smooth solution for the α \alpha -mean curvature flow of codimension two in the classical sense.

This article deals with the existence of high-energy positive solutions for the following coupled Schrödinger system with critical exponent: − Δ u + V 1 ( x ) u = μ 1 u 3 + β u v 2 , x ∈ Ω , − Δ v + V 2 ( x ) v = β u 2 v + μ 2 v 3 , x ∈ Ω , u , v ∈ D 0 1 , 2 ( Ω ) \left\{\begin{array}{l}-\Delta u+{V}_{1}\left(x)u={\mu }_{1}{u}^{3}+\beta u{v}^{2},\hspace{1em}x\in \Omega ,\\ -\Delta v+{V}_{2}\left(x)v=\beta {u}^{2}v+{\mu }_{2}{v}^{3},\hspace{1em}x\in \Omega ,\\ u,v\in {D}_{0}^{1,2}\left(\Omega )\end{array}\right. where Ω ⊂ R 4 \Omega \subset {{\mathbb{R}}}^{4} is an unbounded exterior domain, ∂ Ω ≠ ∅ \partial \Omega \ne \varnothing , R 4 \ Ω {{\mathbb{R}}}^{4}\backslash \Omega is bounded, V 1 , V 2 ∈ L 2 ( Ω ) ∩ L loc ∞ ( Ω ) {V}_{1},{V}_{2}\in {L}^{2}\left(\Omega )\cap {L}_{{\rm{loc}}}^{\infty }\left(\Omega ) are non-negative functions, and μ 1 , μ 2 {\mu }_{1},{\mu }_{2} , and β \beta are the positive constants. Combining variational methods and topological degree theory, we explore the existence of high-energy positive solutions to this system in the case of small perturbations of the source term, i.e., if ∣ V 1 ∣ 2 + ∣ V 2 ∣ 2 {| {V}_{1}| }_{2}+{| {V}_{2}| }_{2} and R 4 \ Ω {{\mathbb{R}}}^{4}\backslash \Omega are small enough in a prescribed sense. It is worth noting that our result still holds in the case Ω = R 4 \Omega ={{\mathbb{R}}}^{4} ; hence, this article can be viewed as an extension of recent results for the Benci-Cerami problem in the framework of coupled Schrödinger systems with critical exponent in R 4 {{\mathbb{R}}}^{4} .

In this article, we investigate the initial value problem for the hyperbolic geometric flow equation, which is derived from hyperbolic geometric flow for Riemannian metric. We find that some of the quadratic nonlinear terms satisfy the null condition by rearranging the equation. Based on the ghost weight and standard energy estimate, the L 1 − L ∞ {L}^{1}-{L}^{\infty } estimate of solutions, and the bootstrap argument, lifespan of smooth solutions is established under the assumption of small data.

In this article, we consider the well-posedness of the local smooth solutions to the physical vacuum free boundary problem of the cylindrical symmetric Euler equations with time-dependent damping − μ ( 1 + t ) λ ρ u -\frac{\mu }{{(1+t)}^{\lambda }}\rho {\bf{u}} , μ > 0 \mu \gt 0 , and λ > 0 \lambda \gt 0 . In this case, the free boundary is moving in the radial direction and the radial velocity will affect the angular velocity, but the axial velocity can be explicitly expressed by ω = ω 0 ( 1 + t ) − μ \omega ={\omega }_{0}{\left(1+t)}^{-\mu } when λ = 1 \lambda =1 and ω = ω 0 e μ 1 − λ [ 1 − ( 1 + t ) 1 − λ ] \omega ={\omega }_{0}{e}^{\tfrac{\mu }{1-\lambda }\left[1-{\left(1+t)}^{1-\lambda }]} when λ ≠ 1 \lambda \ne 1 . Thus, when 0 < λ < 1 0\lt \lambda \lt 1 , ω \omega converges to 0 as t → + ∞ t\to +\infty , and when λ > 1 \lambda \gt 1 , ω \omega converges to ω 0 e μ 1 − λ {\omega }_{0}{e}^{\tfrac{\mu }{1-\lambda }} as t → + ∞ t\to +\infty . This is essentially different from the case of μ = 1 \mu =1 , λ = 0 \lambda =0 in (Meng-Mai-Mei, JDE, 2022), whose axial velocity is only expressed by ω = ω 0 e − t \omega ={\omega }_{0}{e}^{-t} . Moreover, at the moving boundary, the compressible Euler equations with time-dependent damping become a degenerate hyperbolic system. Thanks to the Hardy-type inequality and cut-off functions, we choose a suitable weighted Sobolev spaces to construct a priori estimates to overcome the degeneracy of the system in the Lagrangian coordinates, which confirms the existence of local smooth solution. Gronwall’s inequality guarantees the uniqueness of local smooth solution.

In this article, we deal with the solvability of an infinite system of Volterra-Hammerstein-Stieltjes integral equations in the space of continuous and bounded functions defined on R + {{\mathbb{R}}}_{+} with values in the sequence space l 1 {l}_{1} . We expand a known existence result for such an infinite system of integral equations and prove a theorem for a larger class of infinite systems of integral equations of this type.

In this note, we study the Cauchy problem of the fractional Fokker-Planck equation. We prove that the solution to the Cauchy problem enjoys the analytic smoothing effect with an L 2 {L}^{2} initial datum for positive time, and the evolution of the analytic radius is similar to the heat equation.

This article investigates the fourth-order wave equation with general nonlinear strain term as well as both strong and weak damping terms. By deriving and utilizing energy decay estimate independent of the initial data in the stable manifold, we derive results regarding the continuous dependence of the global solution on initial data for the model equation with such a structure. These conclusions reflect the dissipative nature of the model.

In this work, we develop the classical theory of monotone and pseudomonotone operators in the class of convex-constrained Dirichlet-type problems involving fractional Riesz gradients in bounded and in unbounded domains Ω ⊂ R d \Omega \subset {{\mathbb{R}}}^{d} . We consider the problem of finding u ∈ K s u\in {K}^{s} , such that ∫ R d a ( x , u , D s u ) ⋅ D s ( v − u ) d x + ∫ Ω b ( x , u , D s u ) ( v − u ) d x ≥ 0 \mathop{\int }\limits_{{{\mathbb{R}}}^{d}}{\boldsymbol{a}}\left(x,u,{D}^{s}u)\cdot {D}^{s}\left(v-u){\rm{d}}x+\mathop{\int }\limits_{\Omega }b\left(x,u,{D}^{s}u)\left(v-u){\rm{d}}x\ge 0 for all v ∈ K s v\in {K}^{s} . Here, K s ⊂ Λ 0 s , p ( Ω ) {K}^{s}\subset {\Lambda }_{0}^{s,p}\left(\Omega ) is a non-empty, closed, and convex set of a fractional Sobolev-type space Λ 0 s , p ( Ω ) {\Lambda }_{0}^{s,p}\left(\Omega ) with 0 ≤ s ≤ 1 0\le s\le 1 and 1 < p < ∞ 1\lt p\lt \infty , and D s {D}^{s} denote the distributional Riesz fractional gradient, with two limit cases: D 1 = D {D}^{1}=D represents the classical gradient in the classical Sobolev space Λ 0 1 , p ( Ω ) = W 0 1 , p ( Ω ) {\Lambda }_{0}^{1,p}\left(\Omega )={W}_{0}^{1,p}\left(\Omega ) , and − D 0 = R -{D}^{0}=R denotes the vector-valued Riesz transform within Λ 0 0 , p ( Ω ) = { u ∈ L p ( R d ) : u = 0 a.e. in R d \ Ω } {\Lambda }_{0}^{0,p}\left(\Omega )=\left\{u\in {L}^{p}\left({{\mathbb{R}}}^{d}):\hspace{0.33em}u\left=0\hspace{0.33em}\hspace{0.1em}\text{a.e. in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{d}\backslash \Omega \right\} . We discuss the existence and uniqueness of solutions in this novel framework, and we obtain new results on the continuous dependence, with respect to the fractional parameter s s , of variational solutions corresponding to several classical assumptions on the structural functions a {\boldsymbol{a}} and b b adapted to the fractional framework. We introduce an extension of the Mosco convergence for convex sets K s {K}^{s} with respect to the parameter s s , including the limit cases s = 1 s=1 and s = 0 s=0 , to prove weak or strong convergences of the solutions u s {u}_{s} and their fractional gradients D s u {D}^{s}u , according to different cases. Several applications are illustrated with examples of unilateral problems, including quasi-variational inequalities with constraints of obstacle type u ≥ ψ u\ge \psi and s s -gradient type ∣ D s u ∣ ≤ g | {D}^{s}u| \le g .

This article deals with elliptic systems of the form − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α ( x ) , α = 1 , … , N . -\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}\left(\mathop{\sum }\limits_{\beta =1}^{N}\mathop{\sum }\limits_{j=1}^{n}{a}_{i,j}^{\alpha ,\beta }\left(x,u\left(x))\frac{\partial {u}^{\beta }\left(x)}{\partial {x}_{j}}\right)={f}^{\alpha }\left(x),\hspace{1em}\alpha =1,\ldots ,N. Under ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive local and global boundedness results. Under ellipticity of all the coefficients and “butterfly” support of off-diagonal coefficients, we derive a global boundedness result. This article also considers regularizing effect of a lower-order term.

In this article, we consider the existence and nonexistence of positive solutions for the following critical Neumann problem: − Δ p u + λ u p − 1 = u p * − 1 + f ( x , u ) , x ∈ Ω ; ∣ ∇ u ∣ p − 2 ∇ u ⋅ ν = u p ♯ − 1 , x ∈ ∂ Ω , \left\{\begin{array}{ll}-{\Delta }_{p}u+\lambda {u}^{p-1}={u}^{{p}^{* }-1}+f\left(x,u),& x\in \Omega ;\\ {| \nabla u| }^{p-2}\nabla u\cdot \nu ={u}^{{p}^{\sharp }-1},& x\in \partial \Omega ,\end{array}\right. where Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} is a bounded domain with C 1 {C}^{1} boundary, n ≥ 3 n\ge 3 , p * = n p n − p {p}^{* }=\frac{np}{n-p} , p ♯ = ( n − 1 ) p n − p {p}^{\sharp }=\frac{\left(n-1)p}{n-p} , 1 < p < n 1\lt p\lt n , Δ p u = div ( ∣ ∇ u ∣ p − 2 ∇ u ) {\Delta }_{p}u=\hspace{0.1em}\text{div}\hspace{0.1em}\left({| \nabla u| }^{p-2}\nabla u) is the p p -Laplacian operator, ν \nu denotes the unit outward normal to ∂ Ω \partial \Omega , and f ( x , u ) f\left(x,u) is a subcritical perturbation. Employing the argument established by Brézis and Nirenberg, we establish the existence and nonexistence of positive solutions for aforementioned equation. Additionally, the extremal functions of the Sobolev constants involving p ♯ {p}^{\sharp } and p * {p}^{* } are investigated.

How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. By incorporating these two critical factors, this article proposes an impulsive non-local faecal-oral model with free boundaries. In this framework, non-local diffusion captures the long-range movement of individuals, impulsive effects model periodic disinfection measures, and free boundaries delineate the advancing fronts of infection. We first establish the existence and uniqueness of a non-negative global classical solution to the system. Subsequently, by employing the theory of resolvent positive operators and their perturbations, we analyze the principal eigenvalue, which depends on the infected domain, impulse intensity, and non-local diffusion kernel functions. On the basis of this eigenvalue, we derive sharp threshold criteria distinguishing between disease extinction and propagation. The incorporation of non-local diffusion and impulsive interventions introduces significant mathematical complexities, which are addressed through novel analytical techniques. Finally, numerical simulations are conducted to validate the theoretical results and explore the impact of pulsed interventions. Our findings demonstrate that periodic disinfection effectively mitigates disease transmission, whereas the influence of non-local diffusion varies with the selection of kernel functions.

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second-order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in different spaces, U U and V V , where in general U ⊈ V U\hspace{0.33em} \nsubseteq \hspace{0.33em}V and V ⊈ U V\hspace{0.33em} \nsubseteq \hspace{0.33em}U . The proof is based on the regularization of the dissipation potential using the generalized Moreau-Yosida regularization and a semi-implicit time-discretization scheme, which demonstrates the existence of strong solutions to the regularized problem. The existence of solutions to the original problem is then shown by letting the regularization parameter converge to zero. Furthermore, we establish an energy-dissipation inequality for the solution. We conclude with applications of this abstract theory.

In this article, we consider the initial-boundary value problem for a class of viscoelastic extensible beam equations with logarithmic source term, strong damping term, and weak damping term. We establish the local existence and uniqueness of weak solutions by applying the Faedo-Galerkin method and the contraction mapping principle. Under the framework of potential wells, for subcritical initial energy, we derive the sharp condition for initial data classification, i.e., the global existence and decay estimates of weak solutions in the stable set and the finite-time blow-up of weak solutions in the unstable set. Moreover, we derive sufficient conditions for the finite-time blow-up of weak solutions with negative initial energy and null initial energy. And we obtain an upper bound estimate for the blow-up time.

We study the shape of solutions to certain variational problems in Sobolev spaces with weights that are powers of ∣ x ∣ | x| . In particular, we detect situations when the extremal functions lack symmetry properties such as radial symmetry and antisymmetry. We also prove an isoperimetric inequality for the first nonzero eigenvalue of a weighted Neumann problem.

In this article, we study a nonlocal boundary blow up problem on an interval and obtain the precise asymptotic formula for solutions when the bifurcation parameter in the problem is large.

We consider an integral functional involving the double-phase operator in a metric measure space equipped with a doubling measure. On the basis of suitable hypotheses on the function governing the phase changes, we design a unifying approach to establish Sobolev-Poincaré inequalities. By using such inequalities together with a Caccioppoli type estimate, we obtain the local boundedness and the local Hölder continuity of K K -quasiminimizers of the double-phase energy. Here, we provide a direct approach for establishing the Sobolev-Poincaré inequalities and local Hölder continuity, that is, we do not use neither the method of separation of phases nor auxiliary frozen functional. Finally, we establish the local gradient higher integrability for K K -quasiminimizers of the integral functional. We prove our results via the De Giorgi’s method, imposing that the involved measure is just doubling.

Discrete systems x ( k + 1 ) = A x ( k ) + B x ( k − m ) x\left(k+1)=Ax\left(k)+Bx\left(k-m) , k = 0 , 1 , … k=0,1,\ldots \hspace{0.33em} are analyzed, where m m is a fixed positive integer, A A , B B are constant 3 by 3 matrices and x : { − m , − m + 1 , … } → R 3 x:\left\{-m,-m+1,\ldots \right\}\to {{\mathbb{R}}}^{3} . Assuming that the system is weakly delayed, its general solution is constructed for every case of the Jordan form of the matrix A A . It is shown that, for k ≥ 3 m k\ge 3m or for k ≥ 2 m k\ge 2m , these formulas reduce to simple forms depending on only the three independent parameters generated by the initial values. Formulas connecting these parameters with the initial ones are found. The results are illustrated by examples. Open problems for future research are discussed, and comparisons are given with the previous results.

In this article, we consider the global well-posedness of the initial-boundary value problem to a nonlinear Boussinesq-fluid-structure interaction system, which describes the motion of an incompressible Boussinesq-fluid surrounded by an elastic structure with the heat exchange and is one coupled incompressible Boussinesq equations with the wave equation and heat equation by physical interface boundary conditions. First, the global existence of weak solutions to this problem in two/three dimension is proven by introducing one class of its suitable weak solution and using the compactness method. Then, the uniqueness of the weak solution to this problem in two-dimension is established. Finally, the existence and uniqueness of the global strong and smooth solution to this problem in two dimension is obtained for any smooth large initial data under the assumptions of suitable compatibility conditions by establishing a priori higher order derivative estimates.

We investigate optimal decay estimates of solutions to the 3D Cauchy problem of the rate type viscoelastic fluids. The main novelty of this article involves three aspects: first, we show that the second-order and third-order spatial derivative of the global solution converges to zero at L 2 {L}^{2} -rate ( 1 + t ) − 7 4 {\left(1+t)}^{-\tfrac{7}{4}} and ( 1 + t ) − 9 4 {\left(1+t)}^{-\tfrac{9}{4}} , respectively, which improve previous related decay results. Second, we prove that the decay rate of the m m th order spatial derivative (where m ∈ [ 0 , 3 ] m\in {[}0,3] ) of the extra stress tensor of the field in L 2 {L}^{2} is ( 1 + t ) − 5 4 − m 2 {\left(1+t)}^{-\tfrac{5}{4}-\tfrac{m}{2}} , which is faster than that of the velocity. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates, which coincide with the upper rates. Therefore, our results are precisely optimal in this sense.

The Cauchy problem for the two-component Benjamin-Ono equation is considered. It is shown that this problem is local well-posed in H s ( R ) × H s ( R ) {H}^{s}\left({\mathbb{R}})\times {H}^{s}\left({\mathbb{R}}) for any s > 9 ⁄ 8 s\gt 9/8 . A local smoothing effect in H s ( R ) × H s ( R ) {H}^{s}\left({\mathbb{R}})\times {H}^{s}\left({\mathbb{R}}) , s > 3 ⁄ 2 s\gt 3/2 is also established.

This article is primarily dedicated to the investigation of the initial value problem for the Schrödinger-wave system in dimension four. By employing self-similar transformations in conjunction with the Banach fixed-point theorem, we establish the existence of self-similar blow-up solutions to the four-dimensional Schrödinger-wave system.

In this article, we reformulate the Keller-Segel model by incorporating a nonlinear cellular stimulation term. We then investigate its dynamical behavior by reducing the Keller-Segel model to a dynamical system on a two-dimensional invariant flow. Our analysis confirms the existence of a pulse in bacteria density and a kink front in chemical density. Furthermore, we derive precise conditions for an oscillatory tail and monotonic decay in the front. Two illustrative examples are presented to demonstrate our findings, and the range of wave speeds is determined numerically for each system. Finally, we examine the stability of the pulse by examining the spectrum of the differential operator associated with the governing partial differential equation.

In this work, we investigate a class of p p -Laplacian equations with the Dirichlet boundary condition. Under some new conditions, the existence and multiplicity of nontrivial solutions are proved by means of the variational methods.

When the viscosity coefficients depend on the mass density ρ \rho in the power ρ δ ( δ > 0 ) {\rho }^{\delta }\left(\delta \gt 0) , the existence of smooth solutions with vacuum of the compressible Navier-Stokes equations have received extensive attentions in recent years. In Cao et al. ( Global regular solutions for one-dimensional degenerate compressible Navier-Stokes equations with large data and far field vacuum , SIAM J. Math. Anal. 54 (2022), 4658–4694), when 0 < δ ≤ 1 0\lt \delta \le 1 and γ ≥ δ + 1 2 \gamma \ge \delta +\frac{1}{2} ( γ \gamma : the adiabatic exponent), the authors have established the global existence of a unique regular solution to the Cauchy problem of compressible Navier-Stokes equations with far field vacuum and general large data. In this article, via establishing some new accurate estimates of the effective velocity v v and the velocity u u , we remove the crucial assumption γ ≥ δ + 1 2 \gamma \ge \delta +\frac{1}{2} in Cao-Li-Zhu’s work, and only require that γ > 1 \gamma \gt 1 as 0 < δ ≤ 1 0\lt \delta \le 1 . This improvement allows us to study more physical fluids including air. Moreover, the global existence obtained here does not depend on ρ 0 ∈ L 1 {\rho }_{0}\in {L}^{1} .

Normal form is an important tool in the study of bifurcations but, unlike those for smooth differential systems, normal forms for piecewise-smooth systems have difficulty with the near-identity transformation, which is constructed piecewise but needs to be homeomorphic. Such a difficulty of homeomorphism was encountered when we simultaneously normalized two matrices and obtained the second-order normal form near an equilibrium of FF type for piecewise-smooth monodromic systems. In this article, we establish normal forms for piecewise-smooth monodromic systems near an equilibrium of FF, FP, or PP type. We overcome the difficulty of homeomorphism and generalize both from second-order to any given order and from FF type to all FF, FP, and PP types. This new method can be used to compute Lyapunov constants and discuss degenerate Hopf bifurcations for piecewise-smooth systems.

In this article, the existence of positive weak solutions to a Dirichlet problem driven by the fractional ( p , q ) \left(p,q) -Laplacian and with reaction both weakly singular and non-locally convective (i.e., depending on the distributional Riesz gradient of solutions) is established. Due to the nature of the right-hand side, we address the problem via sub-super solution methods, combined with variational techniques, truncation arguments, as well as fixed point results.

This article is concerned with the sharp forced waves for degenerate diffusion equations in a shifting environment. The degeneracy of diffusion usually causes the forced waves to become sharp. We show that such sharp forced waves exist when the environment improves and propagates with slow speed. To handle the nonautonomous term caused by the shifting environment, we develop a phase transform method to show existence and nonexistence results of nonautonomous and degenerate second-order differential equations.

This study deals with the global boundedness of a classical solution to a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities in n ≤ 3 n\le 3 . Due to the presence of nonlinear sensitivities, obtaining the necessary ‖ w ‖ L ∞ \Vert w{\Vert }_{{L}^{\infty }} estimate for global existence seems difficult because of the strongly coupled structure. To this end, we propose new energy functionals to address this difficulty. Moreover, by constructing Lyapunov functionals, we obtain several results concerning the global stability of classical solutions.

In this article, first, we introduce a new operator ( ∂ t − Δ p ) s u ( z , t ) = C n , s p ∫ − ∞ t ∫ R n ∣ u ( z , t ) − u ( ζ , ϱ ) ∣ p − 2 ( u ( z , t ) − u ( ζ , ϱ ) ) ( t − ϱ ) n 2 + 1 + s p 2 e − ∣ z − ζ ∣ 2 4 ( t − ϱ ) d ζ d ϱ , {\left({\partial }_{t}-{\Delta }_{p})}^{s}u\left(z,t)={C}_{n,sp}\underset{-\infty }{\overset{t}{\int }}\mathop{\int }\limits_{{{\mathbb{R}}}^{n}}\frac{{| u\left(z,t)-u\left(\zeta ,\varrho )| }^{p-2}\left(u\left(z,t)-u\left(\zeta ,\varrho ))}{{\left(t-\varrho )}^{\tfrac{n}{2}+1+\tfrac{sp}{2}}}{e}^{-\tfrac{{| z-\zeta | }^{2}}{4\left(t-\varrho )}}{\rm{d}}\zeta {\rm{d}}\varrho , where 0 < s < 1 0\lt s\lt 1 , 2 ≤ p < ∞ 2\le p\lt \infty . Second, for the purpose of overcoming the challenge brought by the non-locality of the operator ( ∂ t − Δ p ) s {\left({\partial }_{t}-{\Delta }_{p})}^{s} in space-time, the narrow region principle and the maximum principle are studied, and the direct method of moving planes suitable for the operator ( ∂ t − Δ p ) s {\left({\partial }_{t}-{\Delta }_{p})}^{s} is proposed. Based on this, we establish the radial symmetry, monotonicity and non-existence of solutions for the master equations with fractional p p -Laplacian and the Liouville theorem for the homogeneous master equation with fractional p p -Laplacian. Finally, we are confident that the ideas in the proof, as well as those involving perturbation techniques, limit arguments, and Fourier transforms will be useful tools in exploring qualitative properties and the Liouville theorem for solutions to various nonlocal parabolic problems.

In this work, we examine the initial boundary value problem for the thermoelastic system with p p -Laplacian, subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the thermoelastic system with p p -Laplacian. Under subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time. Finally, we provide the lower and upper bounds of the blow-up time for the thermoelastic system with p p -Laplacian.

This article is concerned with the singularity formation of smooth solutions for a nonhomogeneous hyperbolic system arising in magnetohydrodynamics. The system owns four linearly degenerate characteristic fields that influence each other in the relations of second derivatives, making the problem difficult to handle. We develop the characteristic decomposition technique to apply the nonlinear hyperbolic equations with more than two wave characteristics. It is verified that the smooth solution can form a singularity in finite time and the density itself tends to infinity at the blowup point for a special kind of initial data.

In this article, we consider a free boundary problem modeling the growth of a double-layered tumor, which contains quiescent cells and proliferating cells. Compared with that composed of necrotic cells and proliferating cells, the remarkable feature of this model is that quiescent cells actively consume nutrients, introducing an additional nutrient decay term, but at a rate different from that of proliferating cells. The parameter μ \mu in model represents the “aggressiveness” of the tumor. A threshold μ * {\mu }_{* } is determined such that the radially symmetric stationary solution is linearly stable for μ < μ * \mu \lt {\mu }_{* } and unstable for μ > μ * \mu \gt {\mu }_{* } under non-radially symmetric perturbations. Moreover, it is shown that there exist a sequence { μ n } ( ≥ μ * ) \left\{{\mu }_{n}\right\}\left(\ge {\mu }_{* }) and a positive integer n * * ≥ 2 {n}^{* * }\ge 2 , such that a branch of symmetry-breaking stationary solutions bifurcates from the radially symmetric stationary solution for μ = μ n ( even n ≥ n * * ) \mu ={\mu }_{n}\left({\rm{even}}\hspace{0.33em}n\ge {n}^{* * }) .

In this article, we extend the asymptotic method of moving planes to the following logarithmic Laplacian parabolic system: ∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) = f ( t , v ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , ∂ v ∂ t ( η , t ) + ( − △ ) ℒ v ( η , t ) = g ( t , z ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ 0 , ∞ ) , \left\{\begin{array}{ll}\frac{\partial z}{\partial t}\left(\eta ,t)+{\left(-\bigtriangleup )}^{{\mathcal{ {\mathcal L} }}}z\left(\eta ,t)=f\left(t,v\left(\eta ,t)),\hspace{1.0em}& \left(\eta ,t)\in {{\mathbb{B}}}_{1}\left(0)\times \left[0,\infty ),\\ \frac{\partial v}{\partial t}\left(\eta ,t)+{\left(-\bigtriangleup )}^{{\mathcal{ {\mathcal L} }}}v\left(\eta ,t)=g\left(t,z\left(\eta ,t)),\hspace{1.0em}& \left(\eta ,t)\in {{\mathbb{B}}}_{1}\left(0)\times \left[0,\infty ),\\ z\left(\eta ,t)=0,\hspace{1.0em}v\left(\eta ,t)=0,\hspace{1.0em}& \left(\eta ,t)\in {{\mathbb{B}}}_{1}^{c}\left(0)\times \left[0,\infty ),\end{array}\right. where B 1 ( 0 ) {{\mathbb{B}}}_{1}\left(0) denotes the unit ball in R N {{\mathbb{R}}}^{N} . We first establish a maximum principle, an asymptotic narrow region principle, an asymptotic maximum principle, and an asymptotic strong maximum principle for this system. By applying these principles together with the asymptotic method of moving planes, we then prove the asymptotic radial symmetry and monotonicity of positive solutions to the logarithmic Laplacian parabolic system. Finally, we prove a Hopf’s lemma for the parabolic system involving the logarithmic Laplacian ( − Δ ) ℒ {\left(-\Delta )}^{{\mathcal{ {\mathcal L} }}} , as well as an asymptotic Hopf’s lemma applicable to antisymmetric functions.

Given a bounded ( ε , δ ) \left(\varepsilon ,\delta ) -domain Ω ⊆ R N \Omega \subseteq {{\mathbb{R}}}^{N} ( N ≥ 2 N\ge 2 ) whose boundary Γ ≔ ∂ Ω \Gamma := \partial \Omega is a d d -set for d ∈ ( N − p , N ) d\in \left(N-p,N) , we investigate a generalized quasi-linear elliptic boundary value problem governed by the regional fractional p p -Laplacian ( − Δ ) p , Ω s {\left(-\Delta )}_{p,\Omega }^{s} in Ω \Omega , and generalized fractional Wentzell boundary conditions of type C p , s ′ N p ′ ( 1 − s ) u + β ∣ u ∣ q − 2 u + Θ q η u = g on Γ , {C}_{p,s}^{^{\prime} }{{\mathcal{N}}}^{p^{\prime} \left(1-s)}u+\beta {| u| }^{q-2}u+{\Theta }_{q}^{\eta }u=g\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\Gamma , where Θ q η {\Theta }_{q}^{\eta } stands as a nonlocal fractional-type q q -operator on Γ \Gamma (also referred as a Besov q q -map), C p , s ′ N p ′ ( 1 − s ) {C}_{p,s}^{^{\prime} }{{\mathcal{N}}}^{p^{\prime} \left(1-s)} denotes the fractional p p -normal derivative operator in Γ \Gamma , and p , q p,q are two growth exponents acting on the interior and boundary, respectively (which are in general unrelated between each other). We first show that this model equation admits a unique weak solution, which is globally bounded in Ω ¯ \overline{\Omega } . Furthermore, given two distinct weak solution related to this boundary value problem with different data values, we establish a priori L ∞ {L}^{\infty } -estimates for the difference of weak solutions with upper bound depending on the differences of the respective interior and boundary data functions. Additionally, a Weak Comparison Principle is derived, and we conclude by establishing a sort of nonlinear Fredholm alternative related to this generalized elliptic fractional model equation.

In this study, we explore the positive solutions of a nonlinear Choquard equation involving the Green kernel of the fractional operator ( − Δ B N ) − α ⁄ 2 {\left(-{\Delta }_{{{\mathbb{B}}}^{N}})}^{-\alpha /2} in the hyperbolic space, where Δ B N {\Delta }_{{{\mathbb{B}}}^{N}} represents the Laplace-Beltrami operator on B N {{\mathbb{B}}}^{N} , with α ∈ ( 0 , N ) \alpha \in \left(0,N) and N ≥ 3 N\ge 3 . This study is analogous to the Choquard equation in the Euclidean space, which involves the nonlocal Riesz potential operator. We consider the functional setting within the Sobolev space H 1 ( B N ) {H}^{1}\left({{\mathbb{B}}}^{N}) , employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to the fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the problem ( F α , λ {F}_{\alpha ,\lambda } ) in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.

In this article, we are concerned with the existence of normalized solutions for nonlinear Schrödinger equations on compact metric graphs with the nonlinearity f f that is allowed to be mass-supercritical at infinity. By employing a min-max principle for constrained functionals, integrating the monotonicity trick, Morse index and blow-up analysis, we establish the existence of solutions with sufficiently small prescribed mass for a Schrödinger equation on compact metric graphs with a general nonlinearity.

This article deals with a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics u t = Δ u − χ ∇ ⋅ ( u ∇ w ) + u λ 1 − μ 1 u r 1 − 1 + a v + a 1 ∫ Ω u d x + a 2 ∫ Ω v d x , x ∈ Ω , t > 0 , v t = Δ v + ξ ∇ ⋅ ( v ∇ z ) + v λ 2 − μ 2 v r 2 − 1 − b u + b 1 ∫ Ω u d x + b 2 ∫ Ω v d x , x ∈ Ω , t > 0 , τ w t = Δ w − w + v , x ∈ Ω , t > 0 , τ z t = Δ z − z + u , x ∈ Ω , t > 0 \left\{\begin{array}{ll}{u}_{t}=\Delta u-\chi \nabla \cdot \left(u\nabla w)+u\left({\lambda }_{1}-{\mu }_{1}{u}^{{r}_{1}-1}+av+{a}_{1}\mathop{\displaystyle \int }\limits_{\Omega }u{\rm{d}}x+{a}_{2}\mathop{\displaystyle \int }\limits_{\Omega }v{\rm{d}}x\right),& x\in \Omega ,\hspace{1em}t\gt 0,\\ {v}_{t}=\Delta v+\xi \nabla \cdot \left(v\nabla z)+v\left({\lambda }_{2}-{\mu }_{2}{v}^{{r}_{2}-1}-bu+{b}_{1}\mathop{\displaystyle \int }\limits_{\Omega }u{\rm{d}}x+{b}_{2}\mathop{\displaystyle \int }\limits_{\Omega }v{\rm{d}}x\right),& x\in \Omega ,\hspace{1em}t\gt 0,\\ \tau {w}_{t}=\Delta w-w+v,& x\in \Omega ,\hspace{1em}t\gt 0,\\ \tau {z}_{t}=\Delta z-z+u,& x\in \Omega ,\hspace{1em}t\gt 0\end{array}\right. in a bounded and smooth domain Ω ⊂ R n ( n ≥ 1 ) \Omega \subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 1) , where τ ∈ { 0 , 1 } \tau \in \left\{0,1\right\} and the parameters χ , ξ > 0 \chi ,\xi \gt 0 , r 1 , r 2 ≥ 2 {r}_{1},{r}_{2}\ge 2 , λ 1 , λ 2 , μ 1 , μ 2 , a , b > 0 {\lambda }_{1},{\lambda }_{2},{\mu }_{1},{\mu }_{2},a,b\gt 0 , a i , b i ∈ R {a}_{i},{b}_{i}\in {\mathbb{R}} ( i = 1 , 2 ) \left(i=1,2) . We first confirm the global boundedness of classical solutions to the above system under some appropriate parameter conditions. Moreover, by constructing several suitable Lyapunov functionals, it is shown that global solutions converge exponentially or algebraically to the constant stationary solutions, and the corresponding convergence rates are determined.

This study investigates the existence and stability of the contact discontinuity to a piston problem, which is governed by one-dimensional compressible Euler equations under the condition of zero relative velocity between the piston and tube gas. We prove that a contact discontinuity is stable under the assumptions that the perturbations of the initial data and the piston velocity are both small enough in some suitable function spaces. Moreover, using the method developed in this study, similar stability result can be obtained when the contact discontinuity is sealed by two pistons on both sides.