Nonlinear and quasilinear evolution equations: existence, uniqueness, and coieparision of solutions: rate of convergence of the difference method

M. I. Khazan

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 15, Tome 127 (1983), pp. 181-200

Résumé

The Cauchy problem \begin{gather} \frac{du(t)}{dt}=A(t, [u](t))u(t)+f(t),\quad0\leqslant t\leqslant T,\quad u(0)=u_0, \end{gather} in a Banach space $X$ is considered. Here $[u](t)=u|_{[0, t]}$, $f\in L_1(0, T; X)$, and for $t$, $w$ fixed the nonlinear operator $A(t, w)$ is a preusogenerator of a semigroup $e^{sA}$ $(s\geqslant0)$ such that $\|e^{sA}u-e^{sA}v\|\leqslant e^{\omega(r, a)^s}\|u-v\|$ when $u, v, w(r)\in Z_r$ (a ball in $Z\subset X$), $\|Aw(\tau)\|\leqslant a$; conditions on $w$-dependence of $A(t, w)$ allow the “highest order terms” to contain $w$. We prove local and global existence and uniqueness theorems for DS-limit solution of (1), study the differentiability of this solution and it's dependence on $u_0$ and $f$, extending analogous results for the equation $\frac{du(t)}{dt}=A(t)u(t)+f(t)$ with $\omega$-dissipative operators due to Crandall–Pazy, Benilan, Crandall–Evans, Evans, Oharu, Pavel. In quasilinear case our results complement ant generalize the well-known theorem of Kato. Besides that, we obtain estimates of the rate of convergence of difference method and estimated of $\|u(t)-v(t)\|$, where $v$ solves (1) with $A(t, w)$ replaced by $B(t, w)$, these results are new also for equations with dissipative operators.