Abstract This article considers the Cauchy problem for compressible Euler system inR{\bf{R}}with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs toL1(R){L}^{1}\left({\bf{R}})), then the damping is non-effective to the long-time behavior of the solution.

Abstract In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {\int }_{{{\mathbb{R}}}^{N}}| u{| }^{r}{\rm{d}}x=c for given c > 0 . c\gt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

Abstract In this article, we formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H ( x ) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.

Abstract This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

Abstract We study the homogeneous Dirichlet problem for the parabolic equations u t − div ( A ( z , ∣ ∇ u ∣ ) ∇ u ) = F ( z , u , ∇ u ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , {u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with the double phase flux A ( z , ∣ ∇ u ∣ ) ∇ u = ( ∣ ∇ u ∣ p ( z ) − 2 + a ( z ) ∣ ∇ u ∣ q ( z ) − 2 ) ∇ u {\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u and the nonlinear source F F . The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a a , p p , and q q are Lipschitz-continuous, a ( z ) a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p p , and q q satisfy the balance conditions 2 N N + 2 < p − ≤ p ( z ) ≤ q ( z ) < p ( z ) + r ∗ 2 \frac{2N}{N+2}\lt {p}^{-}\le p\left(z)\le q\left(z)\lt p\left(z)+\frac{{r}^{\ast }}{2} with r ∗ = r ∗ ( p − , N ) {r}^{\ast }={r}^{\ast }\left({p}^{-},N) , p − = min Q ¯ T p ( z ) {p}^{-}={\min }_{{\overline{Q}}_{T}}\hspace{0.33em}p\left(z) . It is shown that under suitable conditions on the growth of F ( z , u , ∇ u ) F\left(z,u,\nabla u) with respect to the second and third arguments, the problem has a solution u u with the following properties: u t ∈ L 2 ( Q T ) , ∣ ∇ u ∣ p ( z ) + δ ∈ L 1 ( Q T ) for every 0 ≤ δ < r ∗ , ∣ ∇ u ∣ s ( z ) , a ( z ) ∣ ∇ u ∣ q ( z ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) with s ( z ) = max { 2 , p ( z ) } . \begin{array}{l}{u}_{t}\in {L}^{2}\left({Q}_{T}),\hspace{1.0em}| \nabla u{| }^{p\left(z)+\delta }\in {L}^{1}\left({Q}_{T})\hspace{1.0em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}0\le \delta \lt {r}^{\ast },\\ | \nabla u{| }^{s\left(z)},\hspace{0.33em}a\left(z)| \nabla u{| }^{q\left(z)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))\hspace{1em}{\rm{with}}\hspace{0.33em}s\left(z)=\max \left\{2,p\left(z)\right\}.\end{array} Uniqueness is proven under stronger assumptions on the source F F . The same results are established for the equations with the regularized flux A ( z , ( ε 2 + ∣ ∇ u ∣ 2 ) 1 / 2 ) ∇ u {\mathcal{A}}(z,{({\varepsilon }^{2}+| \nabla u{| }^{2})}^{1\text{/}2})\nabla u , ε > 0 \varepsilon \gt 0 .

Abstract We study the behaviour, when p → + ∞ p\to +\infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{\infty }\left(\Omega ) and we study the behaviour of solutions when p → ∞ p\to \infty .

Abstract In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a nonstandard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solution to this pre-dual problem under some assumptions. We conclude the article by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples illustrate the theoretical findings.

Abstract We prove long-time existence for the negative L 2 {L}^{2} -gradient flow of the p-elastic energy, p ≥ 2 p\ge 2 , with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L 2 {L}^{2} -norm of the normal gradient of the curvature κ \kappa . Long-time existence is proved for the gradient flow of these new energies together with the smooth subconvergence of the evolution equation’s solutions to critical points of the regularized energy in W 2 , p {W}^{2,p} . We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the p-elastic energies. These latter weak solutions also subconverge to critical points of the p-elastic energy.

Abstract In this article, we study discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory. Several examples are provided to illustrate applications of our results.