Well-posedness and parabolicity of the boundary-value problem for systems of partial differential equations

Abstract

The paper examines two-point boundary value problem for systems of linear partial differential equations. Not every system has a well posed two-point boundary value problem. For example, for equation

{$$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial x_1}+i \frac{\partial u(x_1,x_2,t)}{\partial x_2}$$

there are no boundary conditions of the form $au(x_1,x_1,0)+bu(x_1,x_2,T)=\varphi(x_1,x_2)$, under which this boundary value problem will be of a well posed in the Schwartz spaces.

In the work, the conditions for the matrix of the system under which exist well posed boundary-value problem are found in the Schwartz spaces, and the form of these boundary conditions is also indicated. So, for systems with a Hermitian matrix, the boundary value problem with conditions of form

$$u(x,0)+u(x,T)=\varphi(x)$$

will always be of a well posed  in the Schwartz spaces, as well as in the spaces of functions of finite smoothness of power growth. For systems with one spatial variable, it is proved that well posed boundary value problems with the condition $u(x,0)+bu(x,T)=\varphi(x)$ ith positive b always exist. In addition, parabolic boundary value problems with property of increasing smoothness of the solutions are investigated.

Similar results hold for linear partial differential equations. The Helmholtz equation

$$\displaystyle\frac{\partial^2 u(x,t)}{\partial t^2}+\Delta u(x,t)=ku(x,t),$$

is considered as an example. It is not  well posed according to Petrovsky, but for it exist a boundary value problem that is parabolic. For an equation with one spatial variable, sufficient conditions for the second-order equation in time are given, under which parabolic boundary value problems exist.