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.

We consider the initial-value problem for linear delay partial differential equations of the parabolic type. We give a sufficient condition for the stability of the solution of this initial-value problem. We present the stability estimates for the solutions of the first and second order accuracy difference schemes for approximately solving this initial-value problem. We obtain the stability estimates in Hölder norms for the solutions of the initial-value problem of the delay differential and difference equations of the parabolic type.

We find a lower estimation for the projection constant of the projective tensor product $X{\otimes }^{\wedge }Y$ and the injective tensor product $X{\otimes }^{\vee }Y$, we apply this estimation on some previous results, and we also introduce a new concept of the projection constants of operators rather than that defined for Banach spaces.

We give necessary and sufficient conditions for an operator on the space $C\left(T,X\right)$ to be $\left(r,p\right)$-absolutely summing. Also we prove that the injective tensor product of an integral operator and an $\left(r,p\right)$-absolutely summing operator is an $\left(r,p\right)$-absolutely summing operator.