Differential and Integral Equations

Approximation of solutions to non-linear integrodifferential parabolic equations in $L^p$-spaces

Alfredo Lorenzi and Francesca Messina

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Abstract

This paper is concerned with non-linear parabolic integrodifferential equations arising in continuum mechanics, phase-field models, and elsewhere where memory terms are important. More precisely, we will consider a non-linear initial-value problem depending on a small parameter and related to a uniformly elliptic second-order differential operator $A$. The integrodifferential character of our problem is expressed by a convolution term multiplying $A$, the scalar kernel approximating a delta-type function. Consequently, the limit problem reduces to a nonlinear parabolic differential problem involving a multiple of operator $A$. Two different non-linearities, which are locally Lipschitz-continuous in suitable metrics, are considered. The basic aim of the paper consists in determining the rate of convergence of the approximating solutions to the exact one. The results proved in the linear case (cf. [10]) are used here as starting points to solve our problems.

Article information

Source
Differential Integral Equations Volume 20, Number 6 (2007), 693-720.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039433

Mathematical Reviews number (MathSciNet)
MR2323045

Zentralblatt MATH identifier
1212.34174

Subjects
Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Secondary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35K55: Nonlinear parabolic equations 45N05: Abstract integral equations, integral equations in abstract spaces

Citation

Lorenzi, Alfredo; Messina, Francesca. Approximation of solutions to non-linear integrodifferential parabolic equations in $L^p$-spaces. Differential Integral Equations 20 (2007), no. 6, 693--720. https://projecteuclid.org/euclid.die/1356039433.


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