Some nonlinear boundary value problems of evolution equations.

摘要

The variation of constants formula is generalized to adapt to the parabolic integrodifferential equations with nonlinear boundary conditions using Grimmer's resolvent theory of integrodifferential equations and Amann's setting based on the interpolation and extrapolation space techniques and analytic semigroup theory. A heuristic general variation of constants formula for initial-boundary hyperbolic integrodifferential problem is also established. Using these variation of constants formulas as the main tool, we study two problems. First, we obtain the global existence of solutions, the continuity and differentiability of the solutions of parabolic differential and integrodifferential initial-boundary value problems with respect to parameters in the nonlinear boundary conditions. A problem of heat conduction with a nonlinear thermal radiation boundary condition is discussed as an application. We also give an application of the Brezis' abstract convergence theorem based upon the theory of m-accretive operator to a similar concrete example but with more stringent conditions.; Second, we prove that, from the view of wave propagation, each solution of hyperbolic differential or integrodifferential problems with nonhomogeneous (even nonlinear) boundary conditions is a summation of three components. Each component is one of the propagation of initial data, boundary data and forcing term with the same propagation speed. By using the generalized variation of constants formulas, we also obtain that both hyperbolic differential and integrodifferential equations with the same initial value, boundary value, and forcing term, have the same wave propagation behavior including domain of influence and the propagation of singularities.