Boletim da Sociedade Paranaense de Matemática (Jan 2012)
On Wave Equations Without Global a Priori Estimates
Abstract
We investigate the existence and uniqueness of weak solution for amixed problem for wave operator of the type:L(u) = frac{partial^2 u}{partial_t^2}− Delta u + |u|^{rho} − f, rho > 1.The operator is defined for real functions u = u(x,t) and f = f(x,t) where (x, t) in Q a bounded cylinder of R^{n+1}.The nonlinearity |u|^{rho} brings serious difficulties to obtain global a priori estimates by using energy method. The reason is because we have not a definite sign for int_{Omega}|u|^{rho} u dx. To solve this problem we employ techniques of L. Tartar [16], see alsoD.H. Sattinger [12] and we succeed to prove the existence and uniqueness of global weak solution for an initial boundary value problem for the operator L(u), with restriction on the initial data u_0, u_1 and on the function f. With this restriction we are able to apply the compactness method and obtain the unique weak solution.
