A Stability Technique for Evolution Partial Differential by Victor A. Galaktionov

By Victor A. Galaktionov

* Introduces a state of the art approach for the research of the asymptotic habit of strategies to evolution partial differential equations.

* Written by way of verified mathematicians on the vanguard in their box, this mix of soft research and huge program is perfect for a path or seminar in asymptotic research and nonlinear PDEs.

* Well-organized textual content with unique index and bibliography, compatible as a direction textual content or reference volume.

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Extra resources for A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach (Progress in Nonlinear Differential Equations and Their Applications)

Example text

Ixl:5R These conditions generalize the condition of square exponential growth that is well known for the heat equation, and allow for the Cauchy problem to be well posed for nonnegative solutions in a class of optimal initial data. On the other hand, the existence and uniqueness theory extends to data and solutions of any sign when the equation is written in the form Ut = ~(Iulm-I u), in the standard setting Uo ELI (]RN) , where it still generates a semigroup of contractions in LI (]RN), or in classes of growing data.

1 The maps St : uo f-+ u(t) are order-preserving contractions on X = LI(]RN). 15) where holds: 0+ denotes the positive part, max{·, OJ. 15). 15) is called T -contraction and was introduced by Benilan in order to tie together the concepts of contraction and order. We will see that this property extends to other initial- and boundary-value problems for many of the nonlinear heat equations we consider. Actually, the property of T -contraction holds for the heat equation not only when X = L I (]RN), but also in the Lebesgue spaces X = LP (]RN) with any 1 ::: p ::: 00.

The property will also be true for solutions of any sign, but then it does not imply conservation of LI-norm. It is also true for 0 < m < 1 if m ::: (N - 2) / N, but not below that value. Property 4. SOURCE-TYPE SOLUTIONS. 14). However, there is a particular family of solutions that plays a role equivalent in some sense to the fundamental solution for the heat equation. 23) F(xt-/3; C), with parameter C > O. The functions U(x, t; C) were variously called source-type solutions, fundamental solutions, Barenblatt-Pattle solutions, Zel' dovich-Kompaneetz-Barenblatt solutions (ZKB), the last being our preferred option in this text.

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