Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity

S. N. Antontsev ; S. I. Shmarev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 3-19

Résumé

We prove the existence and uniqueness of weak solutions of the Dirichlet problem for the nonlinear degenerate parabolic equations $$ u_{t}=\operatorname{div}(a|u|^{\gamma(x,t)}\nabla u)+\mathbf{b}|u|^{\gamma(x,t)/2}\nabla u-c|u|^{\sigma (x,t)-2}u+d, $$ where $a$, $\mathbf{b}$, $c$, and $d$ are given functions of the arguments $x$, $t$, and $u(x,t)$, and the exponents of nonlinearity $\gamma(x,t)$ and $\sigma(x,t)$ are known measurable and bounded functions of their arguments.