The planar Lp dual Minkowski problem

Science China Mathematics - In this paper we study the Lpq-dual Minkowski problem for the case p < 0 < q. We prove for any positive smooth function f on $$mathbb{S}^{1}$$ , there...     

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.     

Nonlinear biharmonic equations with negative exponents

In this paper, we study global positive C⁴ solutions of the geometrically interesting equation: Δ²u+u−q=0 with q>0 in R³. We will establish several existence and non-existence theorems, including the classification result for q=7 with exactly linear growth condition.     

On some nonlinear equations with critical exponents

Consider the problem

     

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.     

Doubly nonlinear parabolic equations with variable exponents of nonlinearity

In generalized Lebesgue and Sobolev spaces, we consider a mixed problem for a class of parabolic equations with double nonlinearity and nondegenerate minor terms whose exponents of nonlinearity are functions of the space variables. By using the Galerkin method, we establish the conditions of existence of weak solutions of the posed problem.     

The existence of large solutions of semilinear elliptic equations with negative exponents

In this work, we consider semilinear elliptic equations with boundary blow-up whose nonlinearities involve a negative exponent. Combining sub- and super-solution arguments, comparison principles and topological degree theory, we establish the existence of large solutions. Furthermore, we show the existence of a maximal large positive solution.     

Infinite Fujita exponents for nonlinear diffusion equations with localized sources

This paper deals with Cauchy problem to nonlinear diffusion $u_t=\Delta u^m+{\lambda }_1{u}^{p_1}(x,t)+{\lambda }_2{u}^{p_2}(x^1\left(t\right),t)$ with $m\ge 1$, $p_i,{\lambda }_i\ge 0$ ($i=1,2$) and $x^1\left(t\right)$ Hölder continuous. A new phenomenon is observed that the critical Fujita exponent $p_c=+\infty$ whenever ${\lambda }_2>0$. More precisely, the solution blows up under any nontrivial and nonnegative initial data for all $p=max\{p_1,p_2\}\in (1,+\infty )$. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities.     

Global attractors for wave equations with nonlinear interior damping and critical exponents

In this paper we study the global attractors for wave equations with nonlinear interior damping. We prove the existence, regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter, when the growth of the nonlinear terms is critical.     

On critical exponents for semilinear heat equations with nonlinear boundary conditions

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The Cauchy problem for nonlinear elliptic equations

This paper is devoted to investigation of the Cauchy problem for nonlinear equations with a small parameter. They are actually small perturbations of linear elliptic equations in which case the Cauchy problem is ill-posed. To study the Cauchy problem we invoke purely nonlinear methods, such as successive iterations and L^q$L^q$ Sobolev spaces with large q$q$. We also discuss linearisable problems.     

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term is studied. The existence, uniqueness, and blow-up of global solutions of this problem are considered. The local solvability of this problem is also discussed.     

The Dirichlet problem for nonlinear elliptic equations with variable exponent

In this paper we study the Dirichlet problem for nonlinear elliptic equations with variable exponents in Sobolev spaces with variable exponent. We show that for every continuous function $g$ on the boundary there exists a unique continuous extension of $g$.     

Lane-Emden systems with negative exponents

We study the elliptic system