Correct boundary value problems with integral condition in half-space for partial differential equations

Abstract

It is well known that the Cauchy problem is ill posed for partial differential equations with constant coefficients if they do not satisfy the Petrovsky well-posedness condition.   In this work, an integral condition is proposed under which the resulting boundary value problem becomes well-posed in the Schwartz space, as well as in spaces of functions with polynomial growth in spatial variables. The presented integral condition is an integral over a semi-axis with an exponential weight, which ensures the convergence of this integral.   By considering this integral as the Laplace transform of the function $\exp(-t^2/2)$, one can obtain its asymptotics, which allow to estimate the resolving function. This estimate makes it possible to prove a theorem on the well-posedness of the resulting boundary value problem in the Schwartz space, as well as in a scale of functions of finite smoothness with polynomial growth.   Furthermore, the problem is considered for an inhomogeneous differential equation where the right-hand side belongs to the Schwartz space in the spatial variables and is compactly supported in the time variable. For the resulting problem, a Green's function is found and estimated from above. Using this estimate, a theorem on the well-posedness of this problem is proved in the Schwartz space and its dual space.   Examples of equations that are ill posed in the sense of Petrovsky are provided, and specific integral conditions are indicated under which the resulting problems will be well-posed in the Schwartz space. \\ Examples of Petrovsky-incorrect equations are given and specific integral conditions are indicated under which the resulting problems will be correct in the L. Schwartz space. Consider the equation $$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_1^2}-\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_2^2}. $$ For this equation, the Petrovsky conditions are not satisfied, since the polynomial $P(s_1,s_2)=-s_1^2+s_2^2$ is unbounded. But if we add the condition $$\int\limits_0^{\infty} \exp\left(-\displaystyle\frac{t^2}{2}\right) u(x_1,x_2,t)dt=\varphi(x_1,x_2),$$ then which the boundary value problem becomes well-posed in the L.\,Schwartz space, as well as in the scale of Banach spaces.