On Approximation of Solutions of Parabolic Functional Differential Equations in Unbounded Domains
László Simon · Birkhäuser Basel eBooks · 2003
We shall consider initial-boundary value problems for the equation $$D_t u(t,x) - \sum\limits_{j = 1}^n {D_j \left[ {f_j (t,x,u(t,x), abla u(t,x))} \right]} + f_0 (t,x,u(t,x), abla u(t,x)) + h(t,x,\left[ {H(u)} \right](t,x)) = F(t,x),\;(t,x) \in Q_T = (0,T) \times \Omega $$ where Ω⊂Rn is an unbounded domain with sufficiently smooth boundary, H is a linear continuous operator in L P (Q T ), the functions f i , h satisfy the Caratheodory conditions and certain polynomial growth conditions. We shall show that the weak solutions of this problem can be obtained as the limit (as k→∞) similar problems, considered in (0, T) ×Ωk where Ωk⊂Ω are bounded domains with sufficiently smooth boundary, having the property. $$\Omega _k \supset \Omega \cap B_k (B_k = \{ x \in R^n :\left| x \right| < k\} ).$$ Similar statements were proved in [13] for more special equations. There will be also proved a uniqueness theorem and the boundedness of the solutions if some additional conditions are satisfied. We shall prove a theorem on the stabilization of the solutions as t→∞.