Semilinear P.D.E.s with Memory
Some initial and boundary value problems for semilinear partial differential equations containing a continuous memory operator F are studied.
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Outline Some initial and boundary value problems for semilinear partial differential equations containing a continuous memory operator :F are studied. A weak formulation in Sobolev spaces is introduced for the Cauchy-Dirichlet problem associated to the parabolic equation
au at -
Llu + :F(u) = f
in
nx]O, T[,
(1)
where n is an open subset of RN (N ;::: 1), f a given function. Existence of a solution is proved by means of approximation, a priori estimates, passage to the limit. Assuming that :F fulfils a Lipschitz continuity property, existence and uniqueness of the solution are also obtained by expressing the solution of (1) as a fixed point for a contraction mapping. Regularity results and generalizations are presented. In particular, - L1 can be replaced by a (possibly nonlinear) cyclically monotone operator. Existence of maximum and minimum solutions is proved, assuming that the operator -F is order preserving. Existence and uniqueness results are also proved for the initial and boundary value problem associated to the first order hyperbolic equation
au au + F(u) =f at + ax
in ]a, b[x]O, T[.
(2)
Finally we deal with a simple O.D.E., which represents the travelling wave solution of a quasilinear second order hyperbolic equation with hysteresis.
Prerequisites. Acquaintance with the methods of analysis of linear and nonlinear partial differential equations in Sobolev spaces is needed. Definitions of fundamental function spaces are recalled in Sects. XII.I and XII.2. Elements of convex analysis and of the theory of variational inequalities are applied in Sect. X.2. Some results for equations containing order preserving operators (outlined in Sect. XII.6) are used in Sect. X.3. A. Visintin, Differential Models of Hysteresis © Springer-Verlag Berlin Heidelberg 1994
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X. Semilinear P.D.E.s with Memory
X.1 Semilinear Parabolic Equations with Memory This chapter is a continuation of the previous one. In this section we deal with parabolic equations in which a memory operator occurs in the source term. We still assume that c RN (N ~ 1) is an open set of Lipschitz class, fix any T > 0, denote by r the boundary of n, and set Q := nx]O, T[, E := rx]O, T[. We consider the following model equation
n
au &t -
Llu +F(u) =f
in Q,
(1.1)
where F is a continuous memory operator, and f is a given function. Here we do not require F to be rate independent, so applications are not confined to hysteresis effects, although the latter are our main concern. The unknown u may represent the temperature and w a space distribution of thermostats, characterized by continuous hysteresis cycles. More precisely, we assume that at each space-time point (x, t) E Q, w(x, t) depends just on the evolution of u at the same point x in the time interval [0, t] (and on the initial value wo(x), since this information is imbedded in the operator:F). In Sect. Xl7 we shall outline a problem of this sort, issued from biology and chemistry (actually, there the operator F is discontinuous). We assume that F and A are as in Sect. IX. 1, that (IX
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