Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces. (English) Zbl 1082.35085
The paper deals with the parabolic initial-boundary value problem
\[
\begin{aligned} {\partial u\over\partial t}+ A(u)+ g(t,x,u,\nabla u)= f&\quad\text{in }Q,\\ u(x,t)= 0 &\quad\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u_0(x)&\quad\text{in }\Omega,\end{aligned}\tag{1}
\]
where \(\Omega\) is a bounded open set in \(\mathbb{R}^N\) \((N\geq 2)\), \(Q= \Omega\times (0,T)\) with \(T> 0\) and \(A(u)= -\text{div}(a(x, t,u\nabla u))\) with \(a: \Omega\times [0, T]\times\mathbb{R}\times\mathbb{R}^N\to \mathbb{R}^N\) a Leray-Lions type operator satisfying some Orlicz space growth conditions. The authors prove the existence of weak solutions for (1) with the nonlinearity \(g\) having “natural” growth with respect to the gradient.
Reviewer: Vesa Mustonen (Oulu)
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
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