Theoretical and numerical studies of some ill-posed problems in partial differential equations

Smith, Richard
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Three nonlinear initial-boundary value problems are considered. A potential energy well theory applies for solutions of the hyperbolic problem u(,tt) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), u(x,0) = U(x) and u(,t)(x,0) = V(x) in D, when the nonlinearity f is suitably restricted. Here D is a bounded, open, connected subset of R('n); the boundary of D, (PAR-DIFF)D, consists of the disjoint (n-1)-dimen- sional submanifolds (sigma), (SIGMA), and their confluence; (DELTA)(,n) denotes the n-dimensional Laplacian; and (PAR-DIFF)/(PAR-DIFF)n denotes the outward normal derivative. The problem has a global weak solution in each dimen- sion n (GREATERTHEQ) 1 provided U lies in the potential well and the total initial energy is small. The global solution is obtained by expanding in normal modes in terms of the Helmholtz eigenfunctions and the eigenfunctions for a modified Steklov problem. Solutions of the hyperbolic problem which start in a region exterior to the potential well with sufficiently small total initial energy can only exist for a finite time;An analogous existence-nonexistence criterion obtains for glo- bal solutions of the parabolic problem u(,t) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), and u(x,0) = U(x) in D;Let (phi) (ELEM) C('1)(- (INFIN),M) be nonnegative, increasing and satisfy;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);The problem u(,tt) = (DELTA)(,n) u + (epsilon)(phi)(u) in D x (0,T), u = 0 on (PAR-DIFF)D x (0,T), u(x,0) = u(,0)(x) and u(,t)(x,0) = v(,0)(x) in D, has a unique local continuous solution for (epsilon) > 0 sufficiently small in dimensions n = 1,2,3 under appropriate assumptions on (phi), u(,0), v(,0), and (PAR-DIFF)D. The solution u can be continued as long as u < M. A potential well theory is unobtainable for this problem in the Sobolev space H(,0)('1)(D) for n (GREATERTHEQ) 2; however, an a priori inequality for solutions guarantees global existence via energy considerations. Numerical evidence indicates that such an a priori inequality is sometimes satisfied by solutions when n (GREATERTHEQ) 2.