Classes of uniqueness and solvability of mixed problems for certain evolution equations in unbounded domains
The authors consider the mixed problem \[ (1) \quad u|_{t=0}=u_ t|_{t=0}=0, \qquad (2) \quad D_ x^ \alpha u|_{\partial \Omega\times(0,T)}=0 \quad (|\alpha|\leq m) \] for the linear equation \[ u_{tt}+Au_ t+Bu=F(x,t) \tag{3} \] in a cylindrical domain \(\Omega\times(0,T)\), where \(\Omega\) is a domain in \(\mathbb{R}^ n\), \(0<T<\infty\), \[ A=\sum_{|\alpha|, |\beta| \leq m} (- 1)^{|\beta|} D^ \beta (a_{\alpha\beta} D^ \alpha) \qquad\text{and}\qquad B=\sum_{|\alpha|, |\beta| \leq m+1} (-1)^{| \beta|} D^ \beta (b_{\alpha\beta} D^ \alpha) \] are uniformly elliptic operators with non-smooth coefficients \(a_{\alpha\beta}\) and \(b_{\alpha\beta}\). A priori estimates like Saint-Venant's principle for an arbitrary solution of the corresponding homogeneous problem are obtained, and on the basis of these estimates under some growth restrictions for the function \(F(x,t)\) the existence of a generalized solution of the problem (1)--(3) is an unbounded domain is proved.
scientific article; zbMATH DE number 56513
scientific article