Startseite Single peaked traveling wave solutions to a generalized μ-Novikov Equation
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Single peaked traveling wave solutions to a generalized μ-Novikov Equation

  • Byungsoo Moon EMAIL logo
Veröffentlicht/Copyright: 22. Mai 2020
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 10 Heft 1

Abstract

In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

Keywords: generalized μ-Novikov equation; Integrable system; peakons
MSC 2010: 35Q35; 37K45

1 Introduction

We consider the following partial differential equation

mt+k1(u2mx+3uuxm)+k2(2mux+umx)=0, (1.1)

where u(t, x) is a function of time t and a single spatial variable x, and

m:=μ(u)−uxx,μ(u):=∫Su(t,x)dx,

with 𝕊 = ℝ/ℤ which denotes the unit circle on ℝ2. Equation (1.1) can be reduced as μ-Novikov equation [39]

mt+u2mx+3uuxm=0,m=μ(u)−uxx, (1.2)

for k1 = 1 and k2 = 0, and the μ-Camassa-Holm equation [28]

mt+2mux+umx=0,m=μ(u)−uxx, (1.3)

for k1 = 0 and k2 = 1, respectively.

It is known that the Camassa-Holm equation of the following form [2, 20]

mt+2mux+umx=0,m=u−uxx, (1.4)

was proposed as a model for the unidirectional propagation of the shallow water waves over a flat bottom (see also [14, 25]), with u(x, t) representing the height of the water’s free surface in terms of non-dimensional variables. The Camassa-Holm equation (1.4) is completely integrable with a bi-Hamiltonian structure and an infinite number of conservation laws [2, 20], and can be solved by the inverse scattering method [5, 6, 30]. It is of interest to note that the Camassa-Holm equation (1.4) can also be derived by tri-Hamitonian duality from the Korteweg-de Vries equation (a number of additional examples of dual integrable systems derived applying the method of tri-Hamitonian duality can be found in [21, 42]). The Camassa-Holm equation (1.4) has two remarkable features: existence of peakon and multi-peakons [1, 2, 3] and breaking waves, i.e., the wave profile remains bounded while its slope becomes unbounded in finite time [7, 8, 10, 11, 12, 33]. Those peaked solitons were proved to be orbitally stable in the energy space [15, 16] and to be asymptotically stable under the Camassa-Holm flow [38] (see also [26, 27] for other equations). It is worth noting that solutions of this type are not mere abstractizations: the peakons replicate a feature that is characteristic for the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves [9, 13, 48]. Geometrically, the Camassa-Holm equation (1.4) describes the geodesic flows on the Bott-Virasoro group [37, 47] and on the diffeomorphism group of the unit circle under H1 metric [29], respectively. The Camassa-Holm equation (1.4) also arises from a non-stretching invariant planar curve flow in the centro-equiaffine geometry [4, 41]. Well-posedness and wave breaking of the Camassa-Holm equation (1.4) were studied extensively, and many interesting results have been obtained, see [7, 10, 11, 12, 33], for example. The μ-Camassa-Holm equation (1.3) was originally proposed as the model for the evolution of rotators in liquid crystals with an external magnetic field and self interatction [28]. It is interesting to note that this equation is integrable in the sense that it admits the Lax-pair and bi-Hamiltonian structure, and also describes a geodesic flow on the diffeomorphism group of 𝕊 with Hμ(𝕊) metric (which is equivalent to H1(𝕊) metric). Its integrability, well-posedness, blow-up and peakons were discussed in [19, 28].

It is observed that all nonlinear terms in the Camassa-Holm equation (1.4) are quadratic. In contrast to the integrable modified Korteweg-de Vries equation with a cubic nonlinearity, it is of great interest to find integrable Camassa-Holm type equations with cubic or higher-order nonlinearity admitting peakon solitons. Recently, two integrable Camassa-Holm type equtions with cubic nonlinearities have been appeared in literature. One was introduced by Olver and Rosenau [42](called the modified Camassa-Holm equation, see also [18, 21]) by using the tri-Hamiltonian duality approach, which takes the form

mt+[(u2−ux2)m]x=0,m=u−uxx. (1.5)

It was shown that the modified Camassa-Holm equation is integrable with the Lax-pair and the bi-Hamiltonian structure. It has single and multi-peaked traveling waves with a different character than of the Camassa-Holm equation (1.4) [22], and it also has new features of blow-up criterion and wave breaking mechanism. The issue of the stability of peakons for the modified Camassa-Holm equation were investigated in [46]. Like μ-Camassa-Holm equation (1.3), μ-version of the modified Camassa-Holm equation

mt+[(2uμ(u)−ux2)m]x=0,m=μ(u)−uxx (1.6)

was introduced in [44]. Its integrability, wave breaking, existence of peaked traveling waves and their stability were discussed in [34, 44]. The second one is the Novikov equation

mt+u2mx+3uuxm=0,m=u−uxx, (1.7)

which is integrable with the Lax pair [40]. A matrix Lax pair reprsentation to the Novikov equation was founded in [23]. It is also noticed that the Novikov equation admits a bi-Hamiltonian structure [23]. Existence of peaked solitons and multi-peakons for Novikov equation were obtained in [24, 40]. Orbital stability of the peaked solitons to the Novikov equation were discussed in [35]. The μ-Novikov equation (1.2), regarded as a μ-version of the Novikov equation, was introduced first in [39]. The existence of its single peakons was established in [39].

More recently, the following generalized μ-Camassa-Holm equation

mt+k1((2μ(u)u−ux2)m)x+k2(2mux+umx)=0,m=μ(u)−uxx (1.8)

was proposed in [45] as a μ-version of the generalized Camassa-Holm equation with quadradic and cubic nonlinearities

mt+k1((u2−ux2)m)x+k2(2mux+umx)=0,m=u−uxx (1.9)

which was derived by Fokas [18] from the hydrodynamical wave, and can also obtained using the approach of tri-Hamiltonian duality [21, 42] to the bi-Hamiltonian Gardner equation

ut+uxxx+k1u2ux+k2uux=0. (1.10)

Note that the Lax pair of equation (1.9) was obtained in [43]. It was shown in [45] that a scale limit of equation (1.8) yields the following integrable equation

vxt−k1vx2vxx+k2vvxx+12vx2=0, (1.11)

which describes asymptotic dynamics of a short capillarty-gravity wave [17], where v(t, x) denotes the fluid velocity on the surface. Notably, the generalized μ-Camassa-Holm equation (1.8) can be regarded as the integrable model that, in a sense, lies midway between equation (1.9) and its limiting version equation (1.11). It has been known that the generalized μ-Camassa-Holm equation (1.8) is formally integrable in the sense that it admits Lax formulation and bi-Hamiltonian form [45].

The existence of periodic peakons is of interest for nonlinear integrable equations because they are relatively new solitary waves (for most models the solitary waves are quite smooth). Applying the method of tri-Hamiltonian duality[21, 42] to the bi-Hamiltonian representation of the Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV), and Gardner equation, the resulting dual systems, such as Camassa-Holm equation (1.4), the modified Camassa-Holm equation (1.5), and the generalized Camassa-Holm equation (1.9), exhibit nonlinear dispersion, and, in most cases, admit a remarkable variety of non-smooth soliton-like solutions, including peakons, compactons, tipons, rampons, mesaons, and so on [32]. It is known that Camassa-Holm equation (1.4), the modified Camassa-Holm equation (1.5), Novikov equation (1.7), and the generalized Camassa-Holm equation (1.9) [2, 22, 36, 40, 43] admit single peakons of the form

u(t,x)=φc(x−ct)=ae−|x−ct|, (1.12)

where the amplitude a is given by c,3c/2,c, and

−3k2±3k22+83ck14k1withk22+83ck1≥0(k1≠0),

for the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation, and the generalized Camassa-Holm equation, respectively. Their corresponding periodic peakons take the form

u(t,x)=φc(x−ct)=acosh⁡(x−ct−[x−ct]−12)cosh⁡(12), (1.13)

where the amplitude a is also given by c,3ccosh⁡(12)/(1+2cosh2⁡(12),c, and

−3k2cosh⁡(12)±3k22cosh2⁡(12)+43k1c(1+2cosh2⁡(12))2k1(1+2cosh2⁡(12))

with

k22cosh2⁡(12)+43k1c(1+2cosh2⁡(12))≥0,

for the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation, and the generalized Camassa-Holm equation, respectively.

It is worth noting that the periodic peakons of the μ-integrable equation are of a manifestly different character. For example, in [28, 31, 39, 44], the authors showed that the μ-Camassa-Holm equatioin (1.3), μ-Novikov equation (1.2), modified μ-Camassa-Holm equation (1.6), and generalized μ-Camassa Holm equation (1.8) admit periodic peakons of the following form

u(t,x)=φc(x−ct)=aφ(x−ct), (1.14)

where

φ(x)=12x2+2324,x∈−12,12, (1.15)

and φ is extended periodically to the real line, the constant a takes value 12c13,12c13,23c5 and

−13k2±169k22+1200ck150k1with169kx2+1200ck1≥0,

respectively, for the μ-Camassa-Holm equation, μ-Novikov equation, modified μ-Camassa-Holm equation, and generalized μ-Camassa-Holm equation.

Motivated by the recent work [28, 44, 45], the aim of this paper is to investigate the existence of periodic peaked solution of the generalized μ-Novikov equation (1.1). Indeed, in Section 2, we give a short review on the notion of a strong and weak solution of the generalized μ-Novikov equation (1.1) and then show that equation (1.1) admits the periodic peakon, which is given by (1.15) with a replaced by

−6k2±6k22+4ck113k1 (1.16)

where the wave speed c satisfies k22+4ck1≥0.

2 Peaked Traveling Waves

We first introduce the initial value problem of Equation (1.1) on the unit circle 𝕊, that is

mt+k1(u2mx+3uuxm)+k2(2mux+umx)=0,t>0,x∈R,u(0,x)=u0(x),m:=μ(u)−uxx,x∈R,u(t,x+1)=u(t,x),t≥0,x∈R. (2.1)

We then formalize the notion of a strong (or classical) and weak solutions of the Equation (1.1) used throughout this paper.

Definition 2.1

If u ∈ C([0, T), Hs(𝕊)) ∩ C1([0, T), Hs–1(𝕊)) with s > 52 and some T > 0 satisfies (2.1), then u is called a strong solution on [0, T). If u is a strong solution on [0, T) for every T > 0, then it is called a global strong solution.

Note that the inverse operator (μ−∂x2)−1 can be obtained by convolution with the corresponding Green’s function, so that

u=(μ−∂x2)−1m=g∗m, (2.2)

where g is given by [28]

g(x):=12x−[x]−122+2324. (2.3)

Here [x] denote the greatest integer for x∈[−12,12]. Its derivative at x = 0 can be assigned to zero, so one has [31]

gx(x):=0,    x=0x−12,0<x<1. (2.4)

Plugging the formula for m := μ(u) – uxx in terms of u into Equation (1.1) results in the following fully nonlinear partial differential equation:

ut+k1u2ux+32(μ−∂x2)−1∂xμ(u)u2+uux2+12(μ−∂x2)−1ux3+k2uux+(μ−∂x2)−1∂x(2uμ(u)+12ux2)=0. (2.5)

The formulation (2.5) allows us to define the notion of a weak solutions as follows.

Definition 2.2

Given the initial data u0 ∈ W1,3(𝕊), the function u ∈ L∞([0, T); W1,3(𝕊)) is said to be a weak solution to (2.1) if it satisfies the following identity:

∫0T∫Suψt+k113u3ψx−32gx∗μ(u)u2+uux2ψ−12(g∗ux3)ψ+k212u2ψx−gx∗(2uμ(u)+12ux2)ψdxdt+∫Su0(x)ψ(0,x)dx=0,

for any smooth test function ψ(t,x)∈Cc∞([0,T)×S). If u is a weak solution on [0, T) for every T > 0, then it is called a global weak solution.

Our main theorem is in the following.

Theorem 2.1

For any c≥−k224k1, Equation (1.1) admits the peaked periodic-one traveling wave solution uc = ϕc(ξ), ξ = x – ct, where ϕc(ξ) is given by

ϕc(ξ)=a12ξ−122+2324,ξ∈−12,12, (2.6)

where the amplitude

a=−6k2±6k22+4ck113k1,k1≠012c13k2,k1=0,k2≠0 (2.7)

and ϕc(ξ) is extended periodically to the real line with period one.

Proof

Inspired by the forms of periodic peakons for the μ-CH equation [28](See also [44, 45]), we assume that the peaked periodic traveling wave of Equation (1.1) is given by

uc(t,x)=a12ξ−[ξ]−122+2324.

According to Definition 2.2 it is found that uc(t, x) satisfies the following equation

∑j=16Ij:=∫0T∫Suc,tψdxdt+k1∫0T∫Suc2uc,xψdxdt+32k1∫0T∫Sgx∗(μ(uc)uc2+ucuc,x2)ψdxdt+12k1∫0T∫Sg∗(uc,x3)ψdxdt+k2∫0T∫Sucuc,xψdxdt+k2∫0T∫Sgx∗(2μ(uc)uc+12uc,x2)ψdxdt=0, (2.8)

for some T > 0 and every test function ψ(t,x)∈Cc∞([0,T)×S). For any x ∈ 𝕊, one finds that

μ(uc)=a∫0ct12x−ct+122+2324dx+a∫ct112x−ct−122+2324dx=a.

To evaluate Ij, j = 1, ⋯, 6, we need to consider two cases: (i) x > ct, and (ii) x ≤ ct.

For x > ct, we have

μ(uc)uc2+ucuc,x2=a334ξ−124+2312ξ−122+529576,uc2uc,x=a314ξ−125+2324ξ−123+529576ξ−12,2μ(uc)uc+12uc,x2=a232ξ−122+2312anducuc,x=a212ξ−123+2324ξ−12.

On the other hand,

32k1gx∗μ(uc)uc2+ucuc,x2=32k1a3∫Sx−y−[x−y]−1234y−ct−[y−ct]−124+2312y−ct−[y−ct]−122+529576dy=32k1a3∫0ctx−y−1234y−ct+124+2312y−ct+122+529576dy+32k1a3∫ctxx−y−1234y−ct−124+2312y−ct−122+529576dy+32k1a3∫x1x−y+1234y−ct−124+2312y−ct−122+529576dy=k1a3−940ξ−125−2324ξ−123+4871920ξ−12,
12k1g∗(uc,x3)=12a3∫S12x−y−[x−y]−122+2324y−ct−[y−ct]−123dy=12k1a3∫0ct12x−y−122+2324y−ct+123dy+12k1a3∫ctx12x−y−122+2324y−ct−123dy+12k1a3∫x112x−y+122+2324y−ct−123dy=k1a3−140ξ−125+1640ξ−12,

and

k2gx∗2μ(uc)uc+12uc,x2=k2a2∫Sx−y−[x−y]−1232y−ct−[y−ct]−122+2312dy=k2a2∫0ctx−y−1232y−ct+122+2312dy+k2a2∫ctxx−y−1232y−ct−122+2312dy+k2a2∫x1x−y+1232y−ct−122+2312dy=k2a2−12ξ−123+18ξ−12.

It follows that

I1=∫0T∫Suc,tψdxdt=−ca∫0T∫Sξ−12ψ(x,t)dxdt,I2=k1a3∫0T∫S14ξ−125+2324ξ−123+529576ξ−12ψ(x,t)dxdt,I3=k1a3∫0T∫S−940ξ−125−2324ξ−123+4871920ξ−12ψ(x,t)dxdt,I4=k1a3∫0T∫S−140ξ−125+1640ξ−12ψ(x,t)dxdt,I5=k2a2∫0T∫S12ξ−123+2324ξ−12ψ(x,t)dxdt,I6=k2a2∫0T∫S−12ξ−123+18ξ−12ψ(x,t)dxdt.

Plugging above expressions into (2.8), we deduce that for any ψ(t,x)∈Cc∞([0,T)×S)

∑j=16Ij=∫0T∫Saξ−12169144k1a2+1312k2a−cψ(t,x)dxdt.

A similar computation yields for x ≤ ct that

μ(uc)uc2+ucuc,x2=a334ξ+124+2312ξ+122+529576,uc2uc,x=a314ξ+125+2324ξ+123+529576ξ+12,2μ(uc)uc+12uc,x2=a232ξ+122+2312,ucuc,x=a212ξ+123+2324ξ+12,

and

32k1gx∗μ(uc)uc2+ucuc,x2=k1a3−940ξ+125−2324ξ+123+4871920ξ+12,12k1g∗(uc,x3)=k1a3−140ξ+125+1640ξ+12,k2gx∗2μ(uc)uc+12uc,x2=k2a2−12ξ+123+18ξ+12.

This allows us to evaluate

∑j=14Ij=∫0T∫Sξ+12−ac+169144k1a3ψ(t,x)dxdt,I5=k2a2∫0T∫S12ξ+123+2324ξ+12ψ(x,t)dxdt,I6=k2a2∫0T∫S−12ξ+123+18ξ+12ψ(x,t)dxdt.

Hence we arrive at

∑j=16Ij=∫0T∫Saξ+12169144k1a2+1312k2a−cψ(t,x)dxdt.

Since ψ(t, x) is an arbitrary, both cases imply that the parameter a fulfills the equation

169144k1a2+1312k2a−c=0.

Clearly, its solutions are given by which gives (2.7). Thus the theorem is proved.□

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. 2017R1C1B1002336).

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Received: 2019-07-17
Accepted: 2020-03-28
Published Online: 2020-05-22

© 2021 Byungsoo Moon, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Editorial
  2. Editorial to Volume 10 of ANA
  3. Regular Articles
  4. Convergence Results for Elliptic Variational-Hemivariational Inequalities
  5. Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms
  6. Single peaked traveling wave solutions to a generalized μ-Novikov Equation
  7. Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term
  8. On isolated singularities of Kirchhoff equations
  9. On the existence of periodic oscillations for pendulum-type equations
  10. Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation
  11. Nehari-type ground state solutions for a Choquard equation with doubly critical exponents
  12. Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth
  13. The structure of 𝓐-free measures revisited
  14. Solvability of an infinite system of integral equations on the real half-axis
  15. Positive Solutions for Resonant (p, q)-equations with convection
  16. Concentration behavior of semiclassical solutions for Hamiltonian elliptic system
  17. Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation
  18. On variational nonlinear equations with monotone operators
  19. Existence results for nonlinear degenerate elliptic equations with lower order terms
  20. Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation
  21. Ground states and multiple solutions for Hamiltonian elliptic system with gradient term
  22. Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
  23. Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior
  24. Multiple solutions for critical Choquard-Kirchhoff type equations
  25. Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case
  26. Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
  27. A posteriori analysis of the spectral element discretization of a non linear heat equation
  28. Liouville property of fractional Lane-Emden equation in general unbounded domain
  29. Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
  30. On some classes of generalized Schrödinger equations
  31. Variational formulations of steady rotational equatorial waves
  32. On a class of critical elliptic systems in ℝ4
  33. Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold
  34. On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry
  35. Existence, multiplicity and nonexistence results for Kirchhoff type equations
  36. Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems
  37. Convergence analysis for double phase obstacle problems with multivalued convection term
  38. Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent
  39. Boundary value problems associated with singular strongly nonlinear equations with functional terms
  40. Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation
  41. Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth
  42. Concentration results for a magnetic Schrödinger-Poisson system with critical growth
  43. Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian
  44. The concentration-compactness principles for Ws,p(·,·)(ℝN) and application
  45. Regularity for commutators of the local multilinear fractional maximal operators
  46. An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
  47. Local versus nonlocal elliptic equations: short-long range field interactions
  48. Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates
  49. Blowing-up solutions of the time-fractional dispersive equations
  50. Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd
  51. Non-stationary Navier–Stokes equations in 2D power cusp domain
  52. Non-stationary Navier–Stokes equations in 2D power cusp domain
  53. Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
  54. Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
  55. Optimality of Serrin type extension criteria to the Navier-Stokes equations
  56. Fractional Hardy-Sobolev equations with nonhomogeneous terms
  57. New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions
  58. On the set of positive solutions for resonant Robin (p, q)-equations
  59. Solving Composite Fixed Point Problems with Block Updates
  60. Lions-type theorem of the p-Laplacian and applications
  61. Half-space Gaussian symmetrization: applications to semilinear elliptic problems
  62. Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth
  63. Global well-posedness of the full compressible Hall-MHD equations
  64. Σ-Shaped Bifurcation Curves
  65. On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
  66. On singular quasilinear elliptic equations with data measures
  67. On the sub–diffusion fractional initial value problem with time variable order
  68. Partial regularity of stable solutions to the fractional Geľfand-Liouville equation
  69. Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials
  70. Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations
https://doi.org/10.1515/anona-2020-0106
Schlagwörter für diesen Artikel
generalized μ-Novikov equation; Integrable system; peakons
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Editorial
  2. Editorial to Volume 10 of ANA
  3. Regular Articles
  4. Convergence Results for Elliptic Variational-Hemivariational Inequalities
  5. Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms
  6. Single peaked traveling wave solutions to a generalized μ-Novikov Equation
  7. Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term
  8. On isolated singularities of Kirchhoff equations
  9. On the existence of periodic oscillations for pendulum-type equations
  10. Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation
  11. Nehari-type ground state solutions for a Choquard equation with doubly critical exponents
  12. Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth
  13. The structure of 𝓐-free measures revisited
  14. Solvability of an infinite system of integral equations on the real half-axis
  15. Positive Solutions for Resonant (p, q)-equations with convection
  16. Concentration behavior of semiclassical solutions for Hamiltonian elliptic system
  17. Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation
  18. On variational nonlinear equations with monotone operators
  19. Existence results for nonlinear degenerate elliptic equations with lower order terms
  20. Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation
  21. Ground states and multiple solutions for Hamiltonian elliptic system with gradient term
  22. Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
  23. Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior
  24. Multiple solutions for critical Choquard-Kirchhoff type equations
  25. Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case
  26. Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
  27. A posteriori analysis of the spectral element discretization of a non linear heat equation
  28. Liouville property of fractional Lane-Emden equation in general unbounded domain
  29. Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
  30. On some classes of generalized Schrödinger equations
  31. Variational formulations of steady rotational equatorial waves
  32. On a class of critical elliptic systems in ℝ4
  33. Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold
  34. On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry
  35. Existence, multiplicity and nonexistence results for Kirchhoff type equations
  36. Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems
  37. Convergence analysis for double phase obstacle problems with multivalued convection term
  38. Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent
  39. Boundary value problems associated with singular strongly nonlinear equations with functional terms
  40. Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation
  41. Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth
  42. Concentration results for a magnetic Schrödinger-Poisson system with critical growth
  43. Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian
  44. The concentration-compactness principles for Ws,p(·,·)(ℝN) and application
  45. Regularity for commutators of the local multilinear fractional maximal operators
  46. An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
  47. Local versus nonlocal elliptic equations: short-long range field interactions
  48. Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates
  49. Blowing-up solutions of the time-fractional dispersive equations
  50. Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd
  51. Non-stationary Navier–Stokes equations in 2D power cusp domain
  52. Non-stationary Navier–Stokes equations in 2D power cusp domain
  53. Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
  54. Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
  55. Optimality of Serrin type extension criteria to the Navier-Stokes equations
  56. Fractional Hardy-Sobolev equations with nonhomogeneous terms
  57. New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions
  58. On the set of positive solutions for resonant Robin (p, q)-equations
  59. Solving Composite Fixed Point Problems with Block Updates
  60. Lions-type theorem of the p-Laplacian and applications
  61. Half-space Gaussian symmetrization: applications to semilinear elliptic problems
  62. Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth
  63. Global well-posedness of the full compressible Hall-MHD equations
  64. Σ-Shaped Bifurcation Curves
  65. On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
  66. On singular quasilinear elliptic equations with data measures
  67. On the sub–diffusion fractional initial value problem with time variable order
  68. Partial regularity of stable solutions to the fractional Geľfand-Liouville equation
  69. Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials
  70. Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations
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