<b>On Wave Equations Without Global a Priori Estimates</b> - doi: 10.5269/bspm.v30i2.13451
Resumen
We investigate the existence and uniqueness of weak solution for a mixed problem for wave operator of the type:
L(u) =u_{tt}− \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 also D.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.
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