The Exponential Decay Of Energy For Semi-linear Damping Wave Equation

As far as nonlinear hyperbolic equation be concerned, the existence of itsclassical solution may not be guaranteed, let along the decay of its energy. Thispaper starts with a special initial boundary value problem of homogeneous linear equation, proves the existence of classical solution of the problem by usingthe Galerkin's method and consequently yields a energy decay estimate by usingenergy method. On the base of above, We studies the uniqueness of classical solutions and the exponential decay of energy of the initial boundary value problemBy the aid of all the results, we define function space and metric, use Banach's Fixed Point Theorem. Then in a bounded cone_region, we prove theexistence and uniqueness of classical solutions and the exponential decay of energy of the initial boundary value problem for the semi_linear damping waveequationThis paper contents five lemmas and two theorems as well their proves. Thefirst section is introduction which draw forth the problems from two physicalmodels. The second one gives us two principal results. The third one introducesand proves the five lemmas. The fourth is the prove of the two principal results.We will see the emphasis of this paper in the third and fourth sections.