A problem with oblique derivative for some pseudoparabolic equations and equations close to them
In the recent mathematical literature, an equation is called pseudoparabolic provided that it is of the form \[ u_{tt}- Au_t-Bu =f(x,t), \quad u_t- Au_t- Bu=f(x,t), \] with \(A\) and \(B\) some operators of second or higher order in spatial variables. The classical initial-boundary value problems for such equations, both linear and nonlinear, have undergone a rather thorough study over a cylindrical domain \(Q=G \times(0,T)\), \(G\subset \mathbb{R}^n\). In the present article, considering \(A\) and \(B\) to be second-order operators, we study the problem in which some oblique derivative, i.e. a certain first-order differential operator, is given on the lateral boundary. Similar problems are well scrutinized for some classical equations, mainly, for elliptic and parabolic equations and, partly, for hyperbolic equations. However, these problems remain practically unaddressed for nonclassical equations of pseudoparabolic type. The main idea of tackling the above-posed problem is that of transferring the equation under consideration to some new composite-type equation. Earlier, V. N. Vragov succeeded in applying a similar idea to studying a problem with oblique derivative for the wave equation in the case of a boundary operator with constant coefficients and a special spatial domain (see [\textit{V. N. Vragov}, Applications of functional analysis to the equations of mathematical physics, Novosibirsk, 59-66 (1987; Zbl 0741.00038); Boundary value problems for nonclassical equations of mathematical physics, Novosibirsk 100-109 (1987; Zbl 0639.00020)]. In the present article, V. N. Vragov's idea is extended to the case of nonclassical equations with variable coefficients and a spatial domain of arbitrary geometry. We exhibit our approach to the example a pseudoparabolic equation of second order in the variable \(t\). In the article we also indicate some other equations and problems to which this approach applies.
scientific article; zbMATH DE number 1023890
scientific article