Boundary value problems for second-order partial differential equations with operator coefficients

Abstract

Let ${\Omega }_T$ be some bounded simply connected region in ${ℝ}^2$ with $\partial {\Omega }_T={\overline{\Gamma }}_1\cap {\overline{\Gamma }}_2$. We seek a function $u\left(x,t\right)\left(\left(x,t\right)\in {\Omega }_T\right)$ with values in a Hilbert space $H$ which satisfies the equation $ALu\left(x,t\right)=Bu\left(x,t\right)+f\left(x,t,u,u_t\right),\left(x,t\right)\in {\Omega }_T$, where $A\left(x,t\right),B\left(x,t\right)$ are families of linear operators (possibly unbounded) with everywhere dense domain $D$ ($D$ does not depend on $\left(x,t\right)$) in $H$ and $Lu\left(x,t\right)=u_{tt}+a_{11}{u}_{xx}+a_1{u}_t+a_2{u}_x$. The values $u\left(x,t\right);\partial u\left(x,t\right)/\partial n$ are given in ${\Gamma }_1$. This problem is not in general well posed in the sense of Hadamard. We give theorems of uniqueness and stability of the solution of the above problem.