Renormalized solutions of nonlinear parabolic equations with general measure data

Let a bounded open set, N ≥  2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is

Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and G-converging operators

We study the limit as n goes to +∞ of the renormalized solutions u ⁿ to the nonlinear elliptic problems

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

where is a bounded open subset of , , is the so-called Laplace operator, , is a Radon measure with bounded variation on , , , and and belong to the Lorentz spaces , , and , respectively. In particular we prove the existence under the assumptions that , belongs to the Lorentz space , , and is small enough.

     

Semilinear elliptic equations and fixed points

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain , , with boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

     

Global existence for some slightly super-linear parabolic equations with measure data

In this work we study the global existence of a solution to some parabolic problems whose model is

(1)     

Semilinear parabolic equations involving measures and low regularity data

A detailed study of abstract semilinear evolution equations of the form is undertaken, where generates an analytic semigroup and is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.

     

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity

(P)$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+e^u,\hfill & x\in {\mathbf{R}}^N,t>0,\hfill \\ u(x,0)=u_0(x),\hfill & x\in {\mathbf{R}}^N,\hfill \end{array}$

where $u_0$ is a continuous initial function. In this paper, we consider the case where $u_0$ decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for $u_0$ is related to the decay rate of forward self-similar solutions of ${?}_{t}u=\mathrm{\Delta }u+e^u$.     

On exponential stability of linear differential equations with several delays

New explicit conditions of exponential stability are obtained for the nonautonomous linear equation

     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

Explicit exponential stability conditions for linear differential equations with several delays

New explicit conditions of exponential stability are obtained for the nonautonomous linear equation

     

Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term$\{\begin{array}{cc}?\mathrm{\Delta }u+u={\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u}),u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \frac{?u}{?\nu }=0,\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^2$ is a bounded, smooth domain, $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )?\mathbb{N}$, and $q_1,...,q_n\in \mathrm{\Omega }$ are fixed. Given any two non-negative integers $k,l$ with $k+l?1$, it is shown that, for sufficiently small $\epsilon >0$, there exists a solution $u_{\epsilon }$ for which ${\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u})$ asymptotically (i.e. the limit as $\epsilon \to 0$) develops $k+n$ interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function $k(x)=|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}$.     

Problems for elliptic singular equations with a quadratic gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation.     

On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations

For a delay difference equation

     

Nonlinear anisotropic elliptic equations in with variable exponents and locally integrable data

In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.     

Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems

A family of compact and positively invariant sets with uniformly bounded fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the process is constructed. The existence of such a family, called a pullback exponential attractor, is proved for a nonautonomous semilinear abstract parabolic Cauchy problem. Specific examples will be presented in the forthcoming Part II of this work.     

Nonlinear equations with unbounded heat conduction and integrable data

We consider a class of quasi-linear diffusion problems involving a matrix A(t,x,u) which blows up for a finite value m of the unknown u. Stationary and evolution equations are studied for L ¹ data. We focus on the case where the solution u can reach the value m. For such problems we introduce a notion of renormalized solutions and we prove the existence of such solutions.