Stability of Solutions of Quasilinear Parabolic Equations

We bound the difference between solutions $u$ and $v$ of $u_t = a\Delta u+\Div_x f+h$ and $v_t = b\Delta v+\Div_x g+k$ with initial data $\varphi$ and $ \psi$, respectively, by $\Vert u(t,\dott)-v(t,\dott)\Vert_{L^p(E)}\le A_E(t)\Vert \varphi-\psi\Vert_{L^\infty(\R^n)}^{2\rho_p}+ B(t)(\Vert a-b\Vert_{\infty}+ \Vert \nabla_x\cdot f-\nabla_x\cdot g\Vert_{\infty}+ \Vert f_u-g_u\Vert_{\infty} + \Vert h-k\Vert_{\infty})^{\rho_p} \abs{E}^{\eta_p}$. 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 $\rho_p$ and $\eta_p$ 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.