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Nonlinear evolution equations with exponentially decaying memory: existence via time discretisation, uniqueness, and stability. (English) Zbl 1505.47116

The main goal of this paper is to study the existence, uniqueness and stability of solutions of abstract initial value problems defined by \(v'+Av+BKv=f\), where \(A:V_A\rightarrow V'_A\), \(V_A\) is a real reflexive Banach space, \(B:V_B\rightarrow V'_B\), \(V_B\) is a real Hilbert space, \(K\) incorporates a Volterra integral operator in time of convolution type with an exponential decaying kernel. The results are established in a very general setting, assuming more general conditions on the operators \(A\) and \(B\) than those considered until know in the literature (Assumptions A (i)–(iv), Assumptions B (i)–(ii), p. 92). Although the problem studied here is very general, it should be pointed out that it contains as particular cases IBVP that arise is a huge number of applications like in viscoelastic fluids flows, heat flow in materials with memory, non-Fickian diffusion penetrants into viscoelastic materials.
The main results of the present paper are Theorem 4.2 (Existence result for the solution of the IVP (1.1)), Theorem 5.1 (Stability result with respect to perturbations of the problem data), that has as a corollary the uniqueness result, Corollary 5.2.
The main ingredients in the proof of the existence of the solution of (1.1) are the weak convergence in convenient functional spaces of the discrete in time approximations defined by the discrete problem (4.1a)–(4.1c), as well as the integration-by-parts formula (4.9) established in Lemma 4.3 . The weak convergences are deduced taking into account the a priori estimate (4.4) established in the existence and uniqueness result for the time discrete solution – Theorem 4.1.
Lemma 4.3 has also an important role in the proof of the stability result, Theorem 5.1. The paper ends with a stability result with respect to perturbations of the parameter that defines the exponential kernel. The proof of this result requires an integration-by-parts formula similar to (4.9) established in Lemma 4.3. However, the method used to establish (4.9) cannot be applied and the authors obtain the desired formula considering the centered Steklov average.

MSC:

47J35 Nonlinear evolution equations
45K05 Integro-partial differential equations
34K30 Functional-differential equations in abstract spaces
35K90 Abstract parabolic equations
35R09 Integro-partial differential equations
65J08 Numerical solutions to abstract evolution equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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