On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type

Abstract

For linear and semilinear equations of Tricomi type, existence, uniqueness and qualitative properties of weak solutions to the degenerate hyperbolic Goursat problem on characteristic triangles will be established. For the linear problem, a robust L2-based theory will be developed, including well-posedness, elements of a spectral theory, partial regularity results and maximum and comparison principles. For the nonlinear problem, existence of weak solutions with nonlinearities of unlimited polynomial growth at infinity will be proven by combining standard topological methods of nonlinear analysis with the linear theory developed here. For homogeneous supercritical nonlinearities, the uniqueness of the trivial solution in the class of weak solutions will be established by combining suitable Pohožaev-type identities with well tailored mollifying procedures. For the linear problem, the weak existence theory presented here will also be connected to known explicit representation formulas for sufficiently regular solutions with the aid of the partial regularity results. For the nonlinear problem, the question what constitutes critical growth for the problem will be clarified and differences with equations of mixed elliptic–hyperbolic type will be exhibited.