International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 4, Pages 759-769
doi:10.1155/S0161171294001067
Existence of weak solutions for abstract hyperbolic-parabolic equations
Universidade Federal da Paraíba - Campus II - DME - CCT, Campina Grande 58.109-970, Paraíba, Brazil
Received 30 July 1992; Revised 19 April 1993
Copyright © 1994 Marcondes Rodrigues Clark. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper we study the Existence and Uniqueness of solutions for the following Cauchy problem:A2u″(t)+A1u′(t)+A(t)u(t)+M(u(t))=f(t), t∈(0,T) (1)u(0)=u0; A2u′(0)=A212u1; where A1 and A2 are bounded linear operators in a Hilbert space H, {A(t)}0≤t≤T is a family of self-adjoint operators, M is a non-linear map on H and f is a function from (0,T) with values in H.
As an application of problem (1) we consider the following Cauchy problem:k2(x)u″+k1(x)u′+A(t)u+u3=f(t) in Q, (2)u(0)=u0; k2(x)u′(0)=k2(x)12u1 where Q is a cylindrical domain in ℝ4; k1 and k2 are bounded functions defined in an open bounded set Ω⊂ℝ3,A(t)=−∑i,j=1n∂∂xj(aij(x,t)∂∂xi);where aij and a′ij=∂∂tuij are bounded functions on Ω and f is a function from (0,T) with values in L2(Ω).