Blow-up for the Timoshenko-type equation with variable exponents

The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021].     

Singular properties of solutions for a parabolic equation with variable exponents and logarithmic source

This paper deals with a parabolic $p\left(x\right)$-Laplace equation with logarithmic source $u^{q\left(x\right)}logu$. The singular properties of solutions are determined completely by classifying the initial energy. Moreover, we obtain a new extinction rate of solutions, where the order of the extinction rate is greater than the maximum of variable exponent $q\left(x\right)$. This kind of extinction rate could reflect the influence of logarithmic functions on the extinction of solutions more reasonably.     

The Dirichlet problem for nonlinear elliptic equations with variable exponents on Riemannian manifolds

In this paper, after discussing the properties of the Nemytsky operator, we obtain the existence of weak solutions for Dirichlet problems of non-homogeneous p(m)-harmonic equations.     

Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system

     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.     

Existence results for degenerate elliptic equations with critical cone Sobolev exponents

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.     

Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.     

Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations

     

On solutions of anisotropic elliptic equations with variable exponent and measure data

ABSTRACT

The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established. It is proved that the obtained entropy solution is a renormalized solution of the considered problem.     

一类具有周期变指数和凹凸非线性项椭圆型方程解的多重性

研究一类具有周期变指数和凹凸非线性项的椭圆边值问题,借Ekeland分原理和Nehari流形等理论和方法得到解的多重性.     

Existence of ground-state solutions for p-Choquard equations with singular potential and doubly critical exponents

We are concerned with the following Choquard equation: $$-{\mathrm{\Delta }}_{p}u+\frac{A}{{|x|}^{\theta }}{|u|}^{p-2}u=\left(I_{\alpha }\ast F(u)\right)f(u),x\in {\mathbb{R}}^N,$$ where $p\in (1,N)$, $\alpha \in (0,N)$, $\theta \in [0,p)\cup \left(p,\frac{(N-1)p}{p-1}\right)$, $A>0$, ${\mathrm{\Delta }}_p$ is the p-Laplacian, $I_{\alpha }$ is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions-type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions-type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.     

Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity

In this article, new properties of variable exponent Lebesgue and Sobolev spaces are examined. Using these properties we prove the existence of the solution of some parabolic variational inequality.     

Critical exponents for nonlinear diffusion equations with nonlinear boundary sources

This paper is concerned with the large time behavior of solutions to two types of nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q₀ and the critical Fujita exponent qc for the problems considered, and show that q₀=qc for the multi-dimensional porous medium equation and non-Newtonian filtration equation with nonlinear boundary sources. This is quite different from the known results that q₀<qc for the one-dimensional case.     

Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity

We prove the existence and uniqueness of weak solutions of the Dirichlet problem for the nonlinear degenerate parabolic equation

Existence and non-existence of global solutions for uniformly parabolic equations

In this paper, we prove the existence of Fujita-type critical exponents for x-dependent fully nonlinear uniformly parabolic equations of the type $$(*)\quad \partial_{t}u=F(D^{2}u,x)+u^{p}\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ These exponents, which we denote by p(F), determine two intervals for the p values: in ]1,p(F)[, the positive solutions have finite-time blow-up, and in ]p(F), +∞[, global solutions exist. The exponent p(F)?=?1?+?1/α(F) is characterized by the long-time behavior of the solutions of the equation without reaction terms $$\partial_{t}u=F(D^{2}u,x)\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ When F is a x-independent operator and p is the critical exponent, that is, p?=?p(F). We prove as main result of this paper that any non-negative solution to (*) has finite-time blow-up. With this more delicate critical situation together with the results of Meneses and Quaas (J Math Anal Appl 376:514–527, 2011), we completely extend the classical result for the semi-linear problem.     

Weak solutions for a high-order pseudo-parabolic equation with variable exponents

In this paper, the authors investigate the existence and uniqueness of weak solutions of the initial and boundary value problem for a fourth-order pseudo-parabolic equation with variable exponents of non-linearity. Finally, the authors also obtain a long-time behaviour of weak solutions.     

Existence and non-existence of periodic solutions of generalized Lineard equations with damping of no lower bound

This paper establishes criteria for the existence and non-existence of nonzero periodic solutions of the generalized Liénard equationx +f(x,x)x +g(x)=0. The main goal is to study to what extent the dampingf can be small so as to guarantee the existence of nonzero periodic solutions of such a system. With some standard additional assumptions we prove that if for a small ¦x¦, ^± ¦f(x,y)¦^–1 dy=±, then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. Many classical and well-known results can be proved as corollaries to ours.Supported by the National Natural Science Foundation of China.     

Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.