Stability of solutions of quasilinear parabolic equations

Abstract

We bound the difference between solutions $u$ and $v$ of $u_t = aΔu+\Div_x f+h$ and $v_t = bΔv+\Div_x g+k$ with initial data $ϕ$ and $ ψ$, respectively, by $\Vert u(t,\cdot)-v(t,\cdot)\Vert_{L^p(E)}\le A_E(t)\Vert ϕ-ψ\Vert_{L^\infty(\R^n)}^{2ρ_p}+ B(t)(\Vert a-b\Vert_{\infty}+ \Vert \nabla_x\cdot f-\nabla_x\cdot g\Vert_{\infty}+ \Vert f_u-g_u\Vert_{\infty} + \Vert h-k\Vert_{\infty})^{ρ_p} \abs{E}^{η_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\in\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\subset\R^n$ is assumed to be a bounded set, and $ρ_p$ and $η_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.