Classical Solutions of a Mixed Problem with the Zaremba Boundary Condition for a Mildly Quasilinear Wave Equation

Abstract

For a mildly nonlinear wave equation given in the first quadrant, we consider a mixed problem in which we pose the Cauchy conditions on the spatial half-line and the Zaremba condition on the time half-line. Using the method of characteristics, we construct the solution in an implicit analytical form as a solution of some integro-differential equations. The solvability of these equations is studied, as well as the dependence on the initial data and the smoothness of their solutions. For the problem in question, the uniqueness of the solution is proved, and the conditions under which its classical solution exists are established. If the compatibility conditions are not satisfied, then we consider a problem with conjugation conditions. We construct a mild solution if the data are not smooth enough. We apply the obtained mathematical results to solve a problem from combustion theory. Mathematics Subject Classification (2010). Primary 35L20; Secondary 35A02, 35A09, 35D99, 35L71, 35Q79, 80A25.