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Reaction-diffusion equations with infinite delay. (English) Zbl 0836.35158
The results of R. H. Martin and H. L. Smith [Trans. Am. Math. Soc. 321, No. 1, 1-44 (1990; Zbl 0722.35046)] are here generalized to abstract functional differential equations with infinite delay. The equation considered is of the form \[ u(t) = S(t,a) \varphi (0) + \int^t_a T(t,r) F(r,u_r) dr,\;t \geq a, \quad u_a = \varphi \] where \(T\) is a \(C_0\) semigroup and \(S\) is an affine semigroup. (This is required when the theory is applied to systems with mixed boundary conditions.) The first result shows that this abstract equation has a unique maximally defined solution. Next, a number of comparison results are proved, typical of which is the following: if \(v_a^- \leq_B \varphi \leq_B v_a^+\), then the solution \(u\) of the equation satisfies \[ v^- (t) \leq_X u(t) \leq_X v^+ (t), \] where the inequalities are relative to a cone in the appropriate space. The results are finally applied to reaction-diffusion equations with distributed delay and a number of results on existence, invariance and stability are obtained.

35R10 Partial functional-differential equations
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
34K30 Functional-differential equations in abstract spaces