Abstract
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.
Highlights
Suppose ρRemark 2.4 About uniqueness of local weak solution, given by Theorem 2.1, we can apply the same argument of J.L. Lions [10]
The nonlinearity |u|ρ brings serious difficulties to obtain global a priori estimates by using energy method
In this paper we investigate the problem (1.3) but with nonlinearity of the type |u|ρ, ρ > 1, after a remark of N.A
Summary
Remark 2.4 About uniqueness of local weak solution, given by Theorem 2.1, we can apply the same argument of J.L. Lions [10]. We restrict the size of u0 , u1 and f in order to obtain global estimates for approximate solutions um given, system (2.1). These estimates permit us to obtain weak solution for (1.4), defined for all 0 ≤ t < ∞ and x ∈ Ω. Proof: We will obtain global estimates for um(t), solution of the approximate system (2.1), under the assumptions (3.2) and (3.3) for u0 , u1 and f.
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