Existence of a solution for a higher order parabolic equation in unbounded domain, by the method of introducing a parameter

L.Zaręba

In this paper we consider the initial boundary value problem for the equation $u_{tt}+A_{1}u+A_{2}u_{t}+g(u_{t})=f(x,t)$ in an unbounded domain, where $A_{1}$ is a linear elliptic operator of the fourth order and $A_{2}$ is a linear elliptic operator of the second order. Using the method of introducing a parameter we obtain the conditions of the existence of the weak solution for this problem.

1. V. N. Denisov, Some properties of the solutions at $(t\rightarrow +\infty)$ of the iterated heat equation, Diff. Uravn. 28, (1992), № 1, 59--69.

2. V. N. Denisov, Stabilization of the solution of the Cauchy problem for the iterated heat equation, Diff. Uravn. 27, (1991), № 1, 29--42.

3. M. Itano, Some remarks on the Cauchy problem for p-parabolic equations, Hiroshima Math. J. 39(1974), 211--228.

4. J.L. Lions, Quelques methodes de resolution des probl`eme aux limites non lin`eaires, Dunod, Paris, 1969.

5. R. Maecki, E. Zielińska, Contour-integral method applied for solving a certain mixed problem for a parabolic equation of order four, Demon. Math. XV (1982), № 3.

6. M. Mikami, The Cauchy problem for degenerate parabolic equations and Newton polygon, Funkc. Ekvac. 39(1996), 449--468.

7. O. A. Oleinik, E. V. Radkievich, The method of introducing a parameter for the investigation of evolution equations, Uspekhi Mat. Nauk. 203 (1978), № 5, 7--76 (in Russian).

8. P. Rybka, K-H. Hoffmann, Convergence of solutions to the equation of quasi-static approximation of viscoelasticity with capillarity, Journal of Math. Anal. and Appl. 226 (1998), 61--81.

9. K-H. Hoffmann, P. Rybka, On convergence of solutions to the equation of viscoelasticity with capillarity, Comm. Partial Differential Equations. 9--10 (2000), 1845--1890.

10. L. Zareba, The initial boundary value problem for the high order parabolic equation in unbounded domain, Acta Mathematica, Universitatis Iagellonicae. F.XLII (2004), 95--108.

	Existence of a solution for a higher order parabolic equation in unbounded domain, by the method of introducing a parameter
Author	L. Zaręba Lzareba@univ.rzeszow.pl University of Rzeszow,Institute of Mathematics,Rejtana 16A, 35-959 Rzeszow, Poland
Abstract	In this paper we consider the initial boundary value problem for the equation $u_{tt}+A_{1}u+A_{2}u_{t}+g(u_{t})=f(x,t)$ in an unbounded domain, where $A_{1}$ is a linear elliptic operator of the fourth order and $A_{2}$ is a linear elliptic operator of the second order. Using the method of introducing a parameter we obtain the conditions of the existence of the weak solution for this problem.
Keywords	initial value problem, parabolic equation, unbounded domain
DOI	doi:10.30970/ms.28.2.183-190
Reference	1. V. N. Denisov, Some properties of the solutions at $(t\rightarrow +\infty)$ of the iterated heat equation, Diff. Uravn. 28, (1992), № 1, 59--69. 2. V. N. Denisov, Stabilization of the solution of the Cauchy problem for the iterated heat equation, Diff. Uravn. 27, (1991), № 1, 29--42. 3. M. Itano, Some remarks on the Cauchy problem for p-parabolic equations, Hiroshima Math. J. 39(1974), 211--228. 4. J.L. Lions, Quelques methodes de resolution des probl`eme aux limites non lin`eaires, Dunod, Paris, 1969. 5. R. Maecki, E. Zielińska, Contour-integral method applied for solving a certain mixed problem for a parabolic equation of order four, Demon. Math. XV (1982), № 3. 6. M. Mikami, The Cauchy problem for degenerate parabolic equations and Newton polygon, Funkc. Ekvac. 39(1996), 449--468. 7. O. A. Oleinik, E. V. Radkievich, The method of introducing a parameter for the investigation of evolution equations, Uspekhi Mat. Nauk. 203 (1978), № 5, 7--76 (in Russian). 8. P. Rybka, K-H. Hoffmann, Convergence of solutions to the equation of quasi-static approximation of viscoelasticity with capillarity, Journal of Math. Anal. and Appl. 226 (1998), 61--81. 9. K-H. Hoffmann, P. Rybka, On convergence of solutions to the equation of viscoelasticity with capillarity, Comm. Partial Differential Equations. 9--10 (2000), 1845--1890. 10. L. Zareba, The initial boundary value problem for the high order parabolic equation in unbounded domain, Acta Mathematica, Universitatis Iagellonicae. F.XLII (2004), 95--108.
Pages	183-190
Volume	28
Issue	2
Year	2007
Journal	Matematychni Studii
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