Existence of a renormalized solution of nonlinear parabolic equations with general measure data

In this paper we prove the existence of a renormalized solution for nonlinear parabolic equations of the type:$$\displaystyle{\partial b(x,u)\over\partial t} - {\rm div}\Big(a(x,t,\nabla u)\Big)=\mu\qquad \text{in}\ \Omega\times (0,T),$$ where the right handside is a general measure, $b(x,u)$ is anunbounded function of $u$ and $- {\rm div}(a(x,t,\nabla u))$is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in$\nabla u$.

Renormalized solutions for some nonlinear nonhomogeneous elliptic problems with Neumann boundary conditions and right hand side measure

Our aim in this paper is to study the existence of renormalized solution for a class of nonlinear p(x)-Laplace problems with Neumann nonhomogeneous boundary conditions and diuse Radon measure data which does not charge the sets of zero p(.)-capacity

Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data

We give an existence result of a renormalized solution for a classof nonlinear parabolic equations@b(u)/@t div(a(x; t;grad(u))+ H(x; t;ru) = ,where the right side is a general measure, b is a strictly increasing C1-function,div(a(x; t;grad(u)) is a Leray{Lions type operator with growth in grad(u)and H(x; t;grad(u) is a nonlinear lower order term which satisfy the growth condition with respect to grad(u).

On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data

We propose a new definition of renormalized solution to linear equation with self-adjoint operator generating a Markov semigroup and bounded Borel measure on the right-hand side. We give a uniqueness result and study the structure of solutions to truncated equations.

Global radial renormalized solution to a producer–scrounger model with singular sensitivities

This paper is concerned with the parabolic system [Formula: see text] in a bounded ball [Formula: see text] ([Formula: see text]) with [Formula: see text]. Where [Formula: see text] and [Formula: see text]. It is shown that for arbitrarily radially symmetric initial data [Formula: see text], which are nonnegative and suitably regular, the corresponding Neumann initial-boundary problem admits a global renormalized solution, which is moreover smooth in [Formula: see text].

p-Harmonic Functions in ℝN+ with Nonlinear Neumann Boundary Conditions and Measure Data

We study a concept of renormalized solution to the problem\begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases}where {1<p\leq N}, {N\geq 2}, {{\mathbb{R}}^{N}_{+}=\{(x^{\prime},x_{N}):x^{\prime}\in{\mathbb{R}}^{N-1},\,x% _{N}>0\}}, {u_{\nu}} is the normal derivative of u, μ is a bounded Radon measure, and {g:{\mathbb{R}}\rightarrow{\mathbb{R}}} is a continuous function. We prove stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existence of solutions for problems with nonlinear terms of absorption type in both the subcritical and supercritical case. For the problem with source we study the power nonlinearity {g(u)=-u^{q}}, showing existence in the supercritical case, and nonexistence in the subcritical one. We also give a characterization of removable sets when {\mu\equiv 0} and {g(u)=-u^{q}} in the supercritical case.