On the Nonlinear Bi-harmonic Parabolic Equation with data in L-p spaces.
by
Tor A. Kwembe
Department of Mathematics and Computer Science, Chicago State University, 9501 S. King Drive, Chicago, IL 60628

We shall consider the bi-harmonic parabolic equation of the form

u_t(x, t) = - \lambdaѲu(x, t) + \nuÑu^m+1(x, t) ; x in Rⁿ , t > 0

u(x, 0) = f(x) ; x in Rⁿ

where u_t = [(\partial u)/(\partial t)] is the partial derivative of u with respect to time t, Ñ is the Laplacian, Ѳ is the double Laplacian, and \lambda and \nu are positive constants. The presentation centers on establishing the Solonnikov of the fundamental solution \Gamma and those of its spatial and time derivatives. With the use of the embedding theorem, the existence and uniqueness of solutions of the resulting integral equation and hence, of the posed problem in the class of u for which || u||_{p, q}(S_T) < \infty given that T is small. The problem of existence for all time is also considered. We will show that when the initial value f(x) in L^r₁ \cap L^r₂(Rⁿ), r₂ < n < r₁ and the norm || f|| _{L^r1 \cap L^r2(Rⁿ)} = ||f|| _L^r1(Rⁿ) + || f|| _L^r2(Rⁿ) is small enough, then the solution u exists and is unique for all time t.

Date received: March 7, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caky-27.