Stability for Evolution Equations with Variable Growth

Abstract We study the homogeneous Dirichlet problem for evolution p ( x , t )-Laplacian with nonlinear source $$\begin{aligned} u_t-{\text {div}}\left( |\nabla u|^{p(x,t)-2}\nabla u\right) =f(x,t,u),\quad (x,t)\in Q=\Omega \times (0,T). \end{aligned}$$ u t - div | ∇ p ( x , ) 2 = f ∈ Q Ω × 0 T . Here, $$\Omega \subset {\mathbb {R}}^n$$ ⊂ R n is a bounded domain, $$n\ge 2$$ ≥ and $$p(x,\!t)$$ given function $$p(\cdot ):Q\mapsto (\frac{2n}{n+2},p^+]$$ · : ↦ + ] $$p^+<\infty $$ < ∞ . It shown that solution stable respect to perturbations of exponent ), f u initial datum. obtain quantitative estimates on norm difference between two solutions in variable Sobolev space through norms ) data 0), Estimates rate convergence perturbed problems limit are derived.