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# Wydawnictwa / Czasopisma IMPAN / Applicationes Mathematicae / Wszystkie zeszyty

## Mathematical and physical aspects of the initial value problem for a nonlocal model of heat propagation with finite speed

### Tom 40 / 2013

Applicationes Mathematicae 40 (2013), 31-61 MSC: Primary 35L40; Secondary 35E05, 35E15, 80A8. DOI: 10.4064/am40-1-3

#### Streszczenie

Theories of heat predicting a finite speed of propagation of thermal signals have come into existence during the last 50 years. It is worth emphasizing that in contrast to the classical heat theory, these nonclassical theories involve a hyperbolic type heat equation and are based on experiments exhibiting the actual occurrence of wave-type heat transport (so called second sound). This paper presents a new system of equations describing a nonlocal model of heat propagation with finite speed in the three-dimensional space based on Gurtin and Pipkin's approach. We are interested in the physical and mathematical aspects of this new system of equations. First, using the modified Cagniard–de Hoop method we construct a fundamental solution to this system of equations. Next basing on this fundamental solution, we obtain explicit formulae for the solution of the Cauchy problem to this system. Applying the methods of Sobolev space theory, we get an $L^p$-$L^q$ time decay estimate for the solution of the Cauchy problem. For a special form of the source we perform analytical and numerical calculations of the distribution of the temperature for the nonlocal model of heat with finite speed. Some features of the propagation of heat for the nonlocal model are illustrated in a figure together with the comparison of the solution of this model with the solution of the classical heat equation.

#### Autorzy

• Jerzy A. GawineckiInstitute of Mathematics and Cryptology
Faculty of Cybernetics
Military University of Technology
Kaliskiego 2
00-908 Warszawa, Poland
e-mail
• Agnieszka GawineckaCentral Military Bureau
of Design and Technology
Kaliskiego 2
00-908 Warszawa, Poland
• Jarosław ŁazukaInstitute of Mathematics and Cryptology
Faculty of Cybernetics
Military University of Technology
Kaliskiego 2
00-908 Warszawa, Poland
• J. RafaInstitute of Mathematics and Cryptology
Faculty of Cybernetics
Military University of Technology
Kaliskiego 2
00-908 Warszawa, Poland

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