INSTANTANEOUS BLOW-UP FOR NONLINEAR SOBOLEV TYPE EQUATIONS WITH POTENTIALS ON RIEMANNIAN MANIFOLDS

We investigate Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. The first one involves a polynomial nonlinearity and the second one involves a gradient nonlinearity. Namely, we derive sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data, for which the considered problems admit no nontrivial local weak solutions, i.e., an instantaneous blow-up occurs.

Automated summary

This study investigates Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. Sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data are derived to determine whether the considered problems have nontrivial local weak solutions, i.e., an instantaneous blow-up.