Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
-
Wentao Huo
Abstract
This article is concerned with the qualitative properties for the Cauchy problem of a non-Newtonian filtration equation with a reaction source term and volumetric moisture content. On the basis of the slowly decaying behavior of volumetric moisture content, we establish new critical exponents that depend on the ratio of coefficients and exponent of volumetric moisture content. Meantime, under appropriate conditions, we show that the solution globally exists for small enough initial data and blows up in finite time for large enough or any nontrivial initial datum.
1 Introduction
We consider an inhomogeneous quasilinear parabolic equation in the whole dimensional space
subject to the initial condition
where
there exist
where
The quasilinear parabolic equation considered in problems (1.1) and (1.2) is of the p-Laplacian type with the special volumetric moisture content
and thus, the corresponding nonlinear diffusion operator is
We refer to
with
In the past decades, many authors have been devoted to investigating the local well-posedness and qualitative properties of the solutions to
and he studied the Cauchy problem for the classical semilinear parabolic equation
where
Concerning problems (1.1) and (1.2) with volumetric moisture content, without reaction source term, i.e.,
it has been investigated in detail. In particular, depending on the behavior of
For problems (1.1) and (1.2) with volumetric moisture content and a general reaction term, i.e.,
when the behavior of
where
where
initial value
where
then the solution blows up in finite time, in the sense that
Such results have also been generalized to more general initial data, decaying at infinity with a certain rate (cf. [22]).
In this article, continuing with the model considered in the literature [19,40], we consider the Cauchy problems (1.1) and (1.2). However, the key ingredient of the technique in proof is the method of directly constructing the Barenblatt-type super- and sub-solutions, and this article is a first attempt to research the asymptotic behavior of (1.1) and (1.2) involving degenerate
where
Let
At the same time, assume
and define
It is easy to check that
this is consistent with [19, Theorem 1] (see Remark 3.1 for more details). Furthermore, if
Thus, our results are in accordance with those in [12,13,29,32,33]. In addition, for
(See Theorem 5.2.) If
where
(1.5)then the solution blows up for any nontrivial initial data under the additional hypothesis that
In view of (1.3), it can be easily checked that
In particular,
The rest of our article is organized as follows. In Section 2, we introduce some preliminaries related to problems (1.1) and (1.2). In Section 3, we construct suitable super-solution to obtain the existence of global solution. The blow-up results for sufficiently large initial data are proved in Section 4. Finally, for any initial datum, the blow-up of solutions are proved in Section 5.
2 Preliminaries
In this section, we present the definitions of weak solution, weak super- and sub-solutions, comparison of principle and some propositions, which are required in the proof of the main results to problems (1.1) and (1.2).
We first give the definition of the weak solutions of problems (1.1) and (1.2) in different domains.
Definition 2.1
Let
for any nonnegative test function
Similarly, a nonnegative function
For any
When
Definition 2.2
Let
for any nonnegative test function
Meanwhile, a nonnegative function
Proposition 2.1
Assume
Proof
Obviously,
Standard calculations show that
It can be easily verified that
As a result, by applying the standard results (see [7]), problems (2.2)–(2.4) have a nonnegative solution
The proof is completed.□
In addition, the following comparison principle for problems (2.2)–(2.4) holds (cf. [37]).
Proposition 2.2
Assume
Proposition 2.3
Assume
Moreover, u is the minimal solution, that is, the following inequality holds
Proof
Let
Indeed,
Standard calculations show that
It can be easily verified that
It follows from (2.6) and (2.7) that the family
Furthermore, according to the monotone convergence theorem, taking an equal sign in (2.4) and letting
for any nonnegative test function
Next, we aim to prove that
Then letting
As a result,
In a word, the following comparison principles of problems (1.1) and (1.2) can be presented.
Proposition 2.4
Assume
In particular, if
Proof
It is clear that
Letting
Proposition 2.5
Assume
then
Proof
Fix any
then it is easy to know that
Then letting
Remark 2.1
By the similar arguments in [28], one could show that problems (1.1) and (1.2) admits at most one bounded solution, not satisfying any additional condition at infinity, when
In the following, we consider the solution of the following equation:
where
Definition 2.3
Let
for any nonnegative test function
Meanwhile, a nonnegative function
To prove the main results and reader’s convenience, we review the following well-known criterion. Assume
where
Lemma 2.1
Assume
If
(2.13)(2.14)then
Proof
Taking any nonnegative test function
(i) Multiplying (2.13) by
By summing up the previous two inequalities and employing (2.14), we can derive the inequality
Thus, the conclusion holds. The result of (ii) can be obtained by using the same approach. This completes the proof.□
3 Global existence for small initial data
In view of
According to assumptions
Theorem 3.1
Suppose
where
More precisely, if
where
then problems (1.1) and (1.2) admit a global solution
The accurate selection of the parameters
In view of the selection of
Furthermore, from (3.5), we deduce that
Remark 3.1
In [19, Theorem 1], when
For the remainder of this article, we set
To this end, we define the function
where
with
In addition, we denote
Next, we establish the proposition that plays a crucial role in the proof of Theorem 3.1.
Proposition 3.1
Let
Then
Proof
Let
and define
For
By virtue of (3.22), (3.23) and the fact
Then it follows from (3.21) and (3.24) that
By hypothesis
From (3.25) and (3.26), we obtain
Therefore, taking advantage of (3.27) and the definitions of
For each
We aim to find suitable
Indeed, computing the second derivative of
As a result,
which ensures
namely,
which is guaranteed by (1.3), (3.17), and (3.18). Hence, we have proved that
Set
On the other hand, set
In view of
For any
By virtue of (3.33), (3.34), and the fact
From (3.32) and (3.35), we deduce
By applying (3.1), (3.36), and the fact that
Therefore, taking advantage of (3.37) and the definitions of
For each
We aim to find suitable
Indeed, computing the second derivative of
As a result,
which ensures
that is,
which is guaranteed by (3.19) and (3.20). Hence, we have proved that
As a consequence,
Now, observe that
Furthermore, for
In conclusion, set
Remark 3.2
Let
and (1.3) and (3.1) hold. Meanwhile, denote
Lemma 3.1
All the conditions in Remark 3.2 can be fulfilled concurrently.
Proof
We take
In view of
then (3.49) holds. Moreover, making use of (1.3), (3.49), and the fact that
for some
Select
Therefore, from (3.50) and (3.51), inequality (3.48) follows. The proof is completed.□
We are ready to give the proof of the main Theorem 3.1.
Proof of Theorem 3.1
According to Lemma 3.1, we can assume that all the conditions of Remark 3.2 are satisfied. Set
Observe that condition (3.47) implies
In view of
By the assumption
Then (3.54) and (3.55) are direct consequence of (3.44), (3.45), and (3.56).
We now consider conditions (3.19) and (3.20) of Proposition 3.1. Substituting
Obviously, condition (3.57) follows from (3.48) and (3.56). Hence, we can choose suitable
4 Blow-up for large initial data
In this section, we present the blow-up phenomena of the solution to problems (1.1) and (1.2) with sufficiently large initial data.
Theorem 4.1
Suppose
More precisely, we have the following two cases:
(i) Let
where
then the solutions of problems (1.1) and (1.2) blow up and satisfy the bound from below
(ii) Let
Observe that if
Furthermore, in the cases (i) and (ii), from (4.2), we infer that
The accurate selection of the parameters
Remark 4.1
Due to our results concerning any
To show the result of blow-up, we define
where
with
In addition, we define
Next, we establish the proposition that plays a key role in the proof of Theorem 4.1.
Proposition 4.1
Let
Then
Proof
For any
According to hypothesis
Combining (4.17) and (4.18), we obtain
Therefore, taking advantage of (4.19) and the definitions of
For each
We aim to find suitable
To this end, we impose that
with
from which, we obtain
where the coefficient
which implies
Set
On the other hand, set
and define
For any
By virtue of (4.22), (4.23) and the fact
In view of (4.21) and (4.24), we deduce
Then, according to (4.26) and the definitions of
For each
We aim to find suitable
To this end, we impose that
with
from which, we obtain
where the coefficient
which implies
Set
Now, observe that
Moreover, in view of
In conclusion, set
Remark 4.2
Denote
Lemma 4.1
All the conditions in Remark 4.2 can be fulfilled concurrently.
Proof
We are ready to give a proof of the main Theorem 4.1.
Proof of Theorem 4.1
According to Lemma 4.1, we can assume that all the conditions of Remark 4.2 are satisfied. Set
where
Then
Case 1: Let
Case 2: Let
The proof is completed.□
5 Blow-up for any nontrivial initial data
In this section, we deal with a further result with regard to the blow-up of solution to problems (1.1) and (1.2) for any nonnegative and nontrivial initial data
for some
Theorem 5.1
Suppose
More precisely, the bound from (4.2) holds, with
Proof
Since
Without loss of generality, we can suppose that
From inequalities in (5.3), we deduce that
Hence, by Theorem 4.1 and the comparison principle, the result follows. The proof is completed.□
Theorem 5.2
Let assumptions
where
More precisely, when
with
Remark 5.1
Observe that an integral blow-up result of problems (1.1) and (1.2) is shown when
In what follows, let us explain the strategy of the proof of Theorem 5.2. Let
such that
and
for
Combining (5.7) and (5.8), one can see that
Hence,
As a result, it follows from the finite time blow-up property of
Next, we construct a sub-solution of equation (5.5). We define
where
with
Let
and we define
In addition, for
Observe that
and if
We establish the proposition that plays a crucial role in the proof of Theorem 5.2.
Proposition 5.1
Assume
Suppose that
Let
Then there exist
Proof
For any
Now, we define
Hence, (5.21) becomes
where
Now we aim to seek suitable
We observe that it is sufficient that for each
which guarantees
By virtue of the definitions of
By (5.16), (5.18), and (5.19), we have
By applying (5.19), (5.24)–(5.26) can be rewritten as follows:
which reduces to
If (5.27) and (5.28) are verified, then
and
then
then conditions (5.29) and (5.30) are satisfied. As a consequence, by virtue of Lemma 2.1-(ii), we obtain that
On the other hand, for any
Now, we define
Hence, (5.33) becomes
where
Now we aim to seek suitable
we observe that it is sufficient that for each
which guarantees
By virtue of the definitions of
which reduces to
If (5.37) and (5.38) are verified, then
and
Now we select in (5.1)
with
Hence, if
and
then conditions (5.37) and (5.38) are satisfied. Finally, for
then conditions (5.42) and (5.43) are satisfied. Observe that (5.44) is ensured owing to assumptions (5.17) and (5.18). As a consequence, by virtue of Lemma 2.1(ii), we obtain that
Now, observe that
Moreover, for
In conclusion, set
As before, since
Thus, if
and
then (5.6) holds. Furthermore, if
and
then (5.7) follows.
We first show that
then
and
hence (5.47) follows. Furthermore, if
then (5.48) holds. Clearly, for any
then
and
thus (5.48) holds. If
then (5.49) holds. If we select
then (5.53) becomes
Therefore, if
then (5.55) holds. Condition (5.56) is satisfied according to (5.4), for
We are in a position to give a proof of the main Theorem 5.2.
Proof of Theorem 5.2
Let
If
If
Hence,
As a result, it follows from the finite time blow-up property of
Acknowledgments
The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
-
Funding information: This work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2019MA072).
-
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
-
Conflict of interest: The authors declare that they have no competing interests.
-
Ethical approval: Not applicable.
-
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
References
[1] D. Andreucci and A. F. Tedeev, Universal bounds at the blow-up time for nonlinear parabolic equations, Adv. Differential Equ. 10 (2005), no. 1, 89–120. 10.57262/ade/1355867897Suche in Google Scholar
[2] P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. Differ. Equations 68 (1987), 238–252. 10.1016/0022-0396(87)90194-XSuche in Google Scholar
[3] P. Cianci, A. V. Martynenko, and A. F. Tedeev, The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source, Nonlinear Anal. 73 (2010), no. 7, 2310–2323. 10.1016/j.na.2010.06.026Suche in Google Scholar
[4] Y. X. Chen, Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity, Commun. Anal. Mech. 15 (2023), no. 4, 658–694. 10.3934/cam.2023033Suche in Google Scholar
[5] D. Andreucci and A. F. Tedeev, Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds, Milan J. Math. 89 (2021), 295–327. 10.1007/s00032-021-00335-wSuche in Google Scholar
[6] D. Andreucci and A. F. Tedeev, Existence of solutions of degenerate parabolic equations with inhomogeneous density and growing data on manifolds, Nonlinear Anal. 219 (2022), 112818, https://doi.org/10.1016/J.NA.2022.112818. Suche in Google Scholar
[7] E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Suche in Google Scholar
[8] E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1989), 187–224. 10.1090/S0002-9947-1989-0962278-5Suche in Google Scholar
[9] L. F. Dzagoeva and A. F. Tedeev, Asymptotic behavior of the solution of doubly degenerate parabolic equations with inhomogeneous density, Vladikavkaz. Mat. Zh. 24 (2022), no. 3, 78–86. 10.46698/p6936-3163-2954-sSuche in Google Scholar
[10] S. Eidelman, S. Kamin, and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asymptotic Anal. 22 (2000), 349–358. Suche in Google Scholar
[11] H. Fujita, On the blowing up of solutions to the Cauchy problem for ut=Δu+u1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), no. 2, 109–124. Suche in Google Scholar
[12] V. A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita’s exponents, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 517–525. 10.1017/S0308210500028766Suche in Google Scholar
[13] V. A. Galaktionov, Conditions for nonexistence as a whole and localization of the solutions of Cauchyas problem for a class of nonlinear parabolic equations, Zh. Vychisl. Mat. Mat. Fiz. 23 (1985), 1341–1354. Suche in Google Scholar
[14] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Jpn. Acad. 49 (1973), no. 7, 503–505. 10.3792/pja/1195519254Suche in Google Scholar
[15] S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica 28 (1993), 117–120. 10.1007/BF01020323Suche in Google Scholar
[16] S. Kamin, A. Pozio, and A. Tessi, Admissible conditions for parabolic equations degenerating at infinity, St. Petersburg Math. J. 19 (2008), 239–251. 10.1090/S1061-0022-08-00996-5Suche in Google Scholar
[17] X. Li and Z. Y. Xiang, Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation, Commun. Pur. Appl. Anal. 13 (2014), 1465–1480. 10.3934/cpaa.2014.13.1465Suche in Google Scholar
[18] W. Lian, J. Wang, and R. Z. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equations 269 (2020), 4914–4959. 10.1016/j.jde.2020.03.047Suche in Google Scholar
[19] A. V. Martynenko and A. F. Tedeev, On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source, Comput. Math. Math. Phys. 48 (2008), no. 7, 1145–1160. 10.1134/S0965542508070087Suche in Google Scholar
[20] A. V. Martynenko and A. F. Tedeev, Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density, Comput. Math. Math. Phys. 47 (2007), 238–248. 10.1134/S096554250702008XSuche in Google Scholar
[21] A. V. Martynenko and A. F. Tedeev, Regularity of solutions of degenerate parabolic equation with inhomogenious density, Ukr. Mat. Visn. 5 (2008), 116–145. Suche in Google Scholar
[22] A. V. Martynenko, A. F. Tedeev, and V. N. Shramenko, The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source in the class of slowly vanishing initial functions, Izv. Math. 76 (2012), no. 3, 563–580. 10.1070/IM2012v076n03ABEH002595Suche in Google Scholar
[23] A. V. Martynenko, A. F. Tedeev, and V. N. Shramenko, On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes, Ukr. Math. J. 64 (2013), no. 11, 1698–1715. 10.1007/s11253-013-0745-2Suche in Google Scholar
[24] A. V. Martynenko, Global solvability for quasilinear parabolic equation with inhomogeneous density and a source, Appl. Anal. 92 (2013), no. 9, 1863–1888. 10.1080/00036811.2012.708408Suche in Google Scholar
[25] P. Mastrolia, D. D. Monticelli, and F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann. 367 (2017), 929–963. 10.1007/s00208-016-1393-2Suche in Google Scholar
[26] A. D. Pablo, G. Reyes, and A. Sanchez, The Cauchy problem for a nonhomogeneous heat equation with reaction, Discrete Contin. Dyn. Syst., Ser. A 33 (2013), 643–662. 10.3934/dcds.2013.33.643Suche in Google Scholar
[27] R. G. Pinsky, Existence and nonexistence of global solutions for ut=Δu+a(x)up in Rd, J. Differ. Equations 133 (1997), 152–177. 10.1006/jdeq.1996.3196Suche in Google Scholar
[28] F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, J. Evol. Equ. 9 (2009), 429–447. 10.1007/s00028-009-0018-6Suche in Google Scholar
[29] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995. 10.1515/9783110889864Suche in Google Scholar
[30] P. Souplet, Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in RN, J. Funct. Anal. 272 (2017), 2005–2037. 10.1016/j.jfa.2016.09.002Suche in Google Scholar
[31] P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dyn. Syst. 21 (2008), 307–318. 10.3934/dcds.2008.21.307Suche in Google Scholar
[32] Y. W. Qi, Critical exponents of degenerate parabolic equations, Sci. China Ser. A 38 (1995), 1153–1162. Suche in Google Scholar
[33] Y. W. Qi, The global existence and nonuniqueness of a nonlinear degenerate equation, Nonlinear Anal. 31 (1998), 117–136. 10.1016/S0362-546X(96)00298-2Suche in Google Scholar
[34] A. F. Tedeev, Conditions for the time global existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations, Siberian Math. J. 45 (2004), no. 1, 155–164. 10.1023/B:SIMJ.0000013021.66528.b6Suche in Google Scholar
[35] A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. Anal. 86 (2007), no. 6, 755–782. 10.1080/00036810701435711Suche in Google Scholar
[36] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Isr. J. Math. 38 (1981), no. 1–2, 29–40. 10.1007/BF02761845Suche in Google Scholar
[37] Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001. Suche in Google Scholar
[38] R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013), no. 12, 2732–2763. 10.1016/j.jfa.2013.03.010Suche in Google Scholar
[39] H. Y. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech. 15 (2023), no. 2, 132–161. 10.3934/cam.2023008Suche in Google Scholar
[40] J. N. Zhao, On the Cauchy problem and initial traces for the evolution p-Laplacian equations with strongly nonlinear sources, J. Differ. Equations 121 (1995), no. 2, 329–383. 10.1006/jdeq.1995.1132Suche in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Artikel in diesem Heft
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
