Nonlinear evolution equations with noncoercive lower order terms

Abstract

This paper is devoted to study the existence of renormalized solutions of the parabolic Dirichlet equations of prototype: $\left\{\begin{array}{l}\frac{\mathrm{\partial }b(x,u)}{\mathrm{\partial }t}=Lu+f(x,t)\text{in}\mathrm{\Omega }\times (0,T),\\ Lu=\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{p\left(x\right)-2}\mathrm{\nabla }u+c(x,t)|u{|}^{\gamma \left(x\right)}+F)+l(x,t)|u{|}^{\delta \left(x\right)}\\ b(x,u){|}_{t=0}=b(x,u_0(x\left)\right)\text{in}\mathrm{\Omega },\\ u=0\text{on}\mathrm{\partial }\mathrm{\Omega }\times (0,T)\end{array}\right.$where $b(x,s),c(x,t),l(x,t),f(x,t)$ and the exponents of nonlinearities $p\left(x\right),\gamma \left(x\right),\delta \left(x\right)$ are given functions. The main contribution is the proof of an existence result without neither coercivity nor sign condition on non-linearities.

COMMUNICATED BY:

• D. Xu

Keywords:

• Dirichlet parabolic problem
• renormalized solutions
• nonstandard growth conditions
• generalized Gagliardo–Nirenberg inequality

2010 Mathematics Subject Classifications:

• 47A15
• 46A32
• 47D20

Disclosure statement

No potential conflict of interest was reported by the authors.