Abstract:

Let $T$ be a given positive constant. Denote $D :=
\{(x,t): 0<x<1, 0<t<T\}$. We establish stability estimates  of Hölder type for general nonlinear parabolic equations backward in time
\begin{align*}
&u_t=\left(a(x,t) \Phi(u_{x})\right)_x+\gamma uu_x+f(t,u), \quad (x,t) \in D,\\
&u(0,t)=0,\quad u(1,t)=0,\quad 0 \leqslant t \leqslant T,\\
&\|u(\cdot,T)-\chi\|_{L^2(0,1)}\leq \varepsilon,
\end{align*}
under the  condition
\begin{align*}
\max\limits_{(x,t)\in\overline{D}}\{|u_{x}|,|u_{xt}|\}\leqslant E
\end{align*}
with $E$ being some given positive number. Here $\gamma \geq 0$ is a constant, $a(x,t)$ is a smooth function satisfying the conditions
$
0<a_0\leq a(x,t),$ $|a_t(x,t)| \leq M, ~(x,t) \in \overline{D},
$
the function $f$ satisfies the Lipschitz condition
$
\|f(t,\omega_1)-f(t,\omega_2)\|_{L^2(0,1)} \leq k \|\omega_1-\omega_2\|_{L^2(0,1)}
$
and $\chi$ is a given function.