Titlebook: Shock Waves and Reaction—Diffusion Equations; Joel Smoller Textbook 19831st edition Springer-Verlag New York Inc. 1983 Partielle Different

不出名

Abjure

Uniqueness and Energy IntegralsWe shall extend the method of energy integrals to more general second-order (hyperbolic) operators. This “energy” method is a basic technique in the modern theory of partial differential operators, and in the course of our development, we shall establish some interesting and important classical inequalities.

多骨

An Initial-Value Problem for a Hyperbolic EquationWe consider the equation for the homogeneous operator .: . with initial data . We assume that each . is constant, and that the hyperplane . = 0 is non-characteristic with respect to .. Since the normal vector to the surface . = 0 is (1, 0, …, 0), this means . ≠ 0; thus we assume .. = 1.

Amorous

Second-Order Linear Elliptic EquationsSolutions of elliptic equations represent .; i.e., solutions which do not vary with time. They often describe the asymptotic states achieved by solutions of time-dependent problems, as . → ∞. Physically speaking, all the “rough spots” smooth out by the time this steady state is achieved.

dissolution

服從

Holmgren’s Uniqueness Theoremanalytic solution to this problem. Holmgren’s uniqueness theorem denies this possibility. We shall also find this result useful in Chapter 6 where we shall apply it to determine qualitative information on domains of dependence. For this reason, we shall prove a rather general version of the theorem.

審問,審訊

Bifurcation Theorywhen a solution (., .) of (13.1) lies on a “curve” of solutions (., .(.)), at least locally; i.e., for |. — .| < ε. We may also inquire as to when (., .) lies on several solution curves, (., .(.)), (., .(.)),….

Rebate

Comparison Theorems and Monotonicity Methodsabolic equations as . → + ∞. As a second application of the maximum principle, we shall show how it can be used to prove existence theorems. This is the method of “upper” and “l(fā)ower” solutions, the solution being the limit of a monotone iteration scheme, where the monotonicity is a consequence of the maximum principle.

MONY

EXUDE

Characteristics and Initial-Value Problemsoses the existence of a distinguished coordinate, ξ, where the equation ξ = 0 defines the “initial” surface. Of course, as we have seen in the last chapter, one needs some kind of compatibility between the equation and the initial surface. The notion of characteristic serves to classify and make more precise these intuitive ideas.