On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains

TL;DR

This paper investigates the existence and regularity of low regularity solutions to semilinear mixed type equations, extending previous work on degenerate hyperbolic regions to more general mixed type domains.

Contribution

It establishes the existence and regularity of low regularity solutions for semilinear mixed type equations with subcritical growth, in a domain combining hyperbolic and elliptic regions.

Findings

Existence of solutions in mixed type domains.

Solutions have low regularity in Sobolev spaces.

Results extend previous work on degenerate hyperbolic equations.

Abstract

In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation $\p_t^2u-t^{2l-1}\Delta u= f(t,x,u)$ with an initial data $u(0,x)=\varphi(x)\in H^{s}(\Bbb R^n)$ ($0\le s<\f{n}{2}$), where $(t,x)\in\Bbb R \times\Bbb R^n$, $n\ge 2$, $l\in\Bbb N$, $f(t,x,u)$ is $C^1$ smooth in its arguments and has compact support with respect to the variable $x$. Under the assumption of the subcritical growth of $f(t,x,u)$ on $u$, we will show the existence and regularity of the considered solution in the mixed type domain $[-T_0, T_0] \times \R^n $ for some fixed constant $T_0>0$.