Global classical solvability of initial-boundary problems for hyperbolic Lotka-Volterra systems in Sobolev spaces

We consider the global classical solvability of a mixed initial-boundary value problem for semilinear hyperbolic systems with nonlinear reaction of Lotka-Volterra type. The reaction nonlinearity is not globally Lipschitz in L², but has Lipschitz properties depending on an L^¿-norm bound. We reformulate the problem in an abstract setting as a modified Cauchy problem with homogeneous boundary conditions and solve it based on Banach contraction mapping theorem. Based on Sobolev and Moser-type inequalities we prove regularity of the local solutions in Sobolev spaces. We show that global existence of classical solutions holds if a uniform a-priori bound on the L^¿-norm of the solution and boundary term exists.