GLOBAL SOLUTIONS TO THE NONLINEAR HYPERBOLIC PROBLEM

Abstract

We prove the existence and uniqueness of global regular solutions to the mixed problem for the nonlinear hyperbolic equation with nonlinear damping. Utt a (u) u + I u t IP Ut = f (x .t ) in (O, 1) x (O, T ) = Q, u (o.. ) = u (l,t ) = 0, u(x,O) = Uo(x), ut (x,O) = ui (x), Where a (u ) ~ ao > 0, p > 1.No restrictions on a size of uo, ui, f are imposed. It is well-known that quasilinear hyperbolic equations, generally speaking, do not have regular solutions for all t >0_Their solutions can blow up at a finite period of time. See examples of such singularities in [1, 3]. On the other hand,it was observed that adding a linear damping to the nonlinear hyperbolic equations one can expect the existence of global regular solutions provided initial conditions and right-hand side have sufficiently small appropriate norms, [2]. Moreover, in [3] was shown that the presence of the nonlinear damping allows to prove the existence of regular solutions for the equation without restrictions on a size ofthe initial data and f. Later, using the idea of [3], we proved in [4] the existence of regular solutions for the damped Carrier equation. Utt -M ~u(t)12)L1U + alu, IP u, = f without smallness conditions for the initial data and f . • Supported by CNP-Brazil as Visiting Profesor as the State University ofMaringa. Here, we continue to exploit this idea and consider the following nonlinear mixed problem.