An existence result of a renormalized solution for a class of non- linear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.

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Existence results for a class of nonlinear parabolic equations in orlicz spaces

Renormalized Solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz Spaces

We shall give the proof of existence results for the entropy solutions of the following nonlinear parabolic problem 


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where A is a Leray–Lions operator having a growth not necessarily of polynomial type. The lower order term $\Phi $ :$\Omega \times (0,T)\times \mathbb {R}\rightarrow \mathbb {R}^N$ is a Caratheodory function, for a.e. $(x,t)\in Q_T$ and for all $s\in \mathbb {R}$, satisfying only a growth condition and the right hand side f belongs to $L^1(Q_T)$.

Entropy solutions for a nonlinear parabolic problems with lower order term in Orlicz spaces

The paper deals with the existence of solutions of some unilateral problems in the framework of Orlicz spaces.

On some class of nonlinear parabolic inequalities in Orlicz spaces

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, $$\begin{aligned} \left\{ \begin{array}{ll} -Au= \rho (u)|\nabla \varphi |^2 &{}\quad \mathrm{in}\ \Omega , \\ \mathop {\mathrm{div}}\nolimits (\rho (u)\nabla \varphi ) =0&{}\quad \mathrm{in}\ \Omega , \\ \varphi =\varphi _0 &{}\quad \hbox {on } \partial \Omega , \\ u=0&{}\quad \mathrm{on}\, {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset {\mathbb {R}}^d$$, $$d\ge 2$$ and $$\displaystyle Au=-\mathop {\mathrm{div}}\nolimits a(x,u,\nabla u)$$ is a Leray–Lions operator defined on $$W_0^{1}L_M(\Omega )$$, M is a N-function which does not have to satisfy a $$\Delta _2$$ condition. Therefore, we work with generalized Orlicz–Sobolev spaces which are not necessarily reflexive. The function $$\varphi _0$$ is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces.

Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces

Strongly nonlinear parabolic equations with natural growth terms and data in Orlicz spaces.

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Strongly nonlinear parabolic equations with natural growth terms and data in Orlicz spaces.