Publisher Summary This chapter discusses the formulation of the main notions and results in the theory of monotone type operators and their application to (possibly nonlinear) parabolic differential equations and parabolic functional differential equations. The abstract Cauchy problem is considered in the chapter for first order evolution equations in a finite interval. The main results on the existence, uniqueness, and continuous dependence of the weak solutions of higher order nonlinear parabolic differential equations are also provided in the chapter. The chapter includes higher order nonlinear functional parabolic equations, where only the lower order terms contain functional dependence. It also details second order nonlinear parabolic functional differential equations, where also the main part contains functional dependence. It discusses the existence and qualitative properties of the solutions of parabolic functional differential equations in (0, ∞) and elaborates further applications of monotone type operators, for example, to the systems of functional parabolic equations. Several examples are also given in the chapter for the “general” results.

Chapter 6 Application of Monotone Type Operators to Parabolic and Functional Parabolic PDE's

Application of monotone type operators to nonlinear PDE's

We shall consider initial-boundary value problems for the equation 

$$D_t u(t,x) - \sum\limits_{j = 1}^n {D_j \left[ {f_j (t,x,u(t,x),\nabla u(t,x))} \right]} + f_0 (t,x,u(t,x),\nabla u(t,x)) + h(t,x,\left[ {H(u)} \right](t,x)) = F(t,x),\;(t,x) \in Q_T = (0,T) \times \Omega $$

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→∞.

On Approximation of Solutions of Parabolic Functional Differential Equations in Unbounded Domains

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On Approximation of Solutions of Parabolic Functional Differential Equations in Unbounded Domains