×

Cauchy problems of semilinear pseudo-parabolic equations. (English) Zbl 1179.35178

The Cauchy problems \(u(x,0)=u_0 (x)\), \(x\in \mathbb R^n\), for semilinear Sobolev type equation \((u-\kappa \Delta u)_t = \Delta u +u^p\) in \(\mathbb R^n\times\mathbb R_+\) is under consideration. Here parameters \(k\), \(p\in\mathbb R_+\) and function \(u_0 (x)\) is nonnegative and appropriately smooth. Authors prove existence and uniquess of mild solution to this problems.
Main results of this paper are the following: (i) there exist global solution for each initial data in the case \(0 < p \leq 1\), while there exists at least one initial data such that the solution blows up in a finite time in the case \(p>1\); (ii) any nontrivial solution blows up in a finite time in the case \(1<p \leq 1+2/n\), while there exist at least one nontrivial global solution in the case \(p>1+2/n\).

MSC:

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35B44 Blow-up in context of PDEs
35K58 Semilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andreucci, D.; Cirmi, G. R.; Leonardi, S.; Tedeev, A. F., Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174, 253-288 (2001) · Zbl 0987.35087
[2] Barenblat, G.; Zheltov, I.; Kochiva, I., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24, 5, 1286-1303 (1960) · Zbl 0104.21702
[3] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272, 1220, 47-78 (1972) · Zbl 0229.35013
[4] Brill, H., A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24, 3, 412-425 (1977) · Zbl 0346.34046
[5] David, C.; Jet, W., Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69, 2, 411-418 (1979) · Zbl 0409.35050
[6] Deng, K.; Levine, H. A., The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243, 1, 85-126 (2000) · Zbl 0942.35025
[7] Favini, A.; Yagi, A., Degenerate Differential Equations in Banach Spaces, Pure Appl. Math. (N. Y.), vol. 215 (1999), Marcel Dekker: Marcel Dekker New York · Zbl 0913.34001
[8] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(\partial u / \partial t = \Delta u + u^{1 + \alpha}\), J. Fac. Sci. Univ. Tokyo Sect. I, 13, 109-124 (1966) · Zbl 0163.34002
[9] Gopala Rao, V. R.; Ting, T. W., Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49, 57-78 (1972/73) · Zbl 0255.35049
[10] Hayakawa, K., On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49, 503-525 (1973) · Zbl 0281.35039
[11] Kaikina, E. I.; Naumkin, P. I.; Shishmarev, I. A., The Cauchy problem for a Sobolev type equation with power like nonlinearity, Izv. Math., 69, 1, 59-111 (2005) · Zbl 1079.35024
[12] Karch, G., Asymptotic behaviour of solutions to some pesudoparabolic equations, Math. Methods Appl. Sci., 20, 3, 271-289 (1997) · Zbl 0869.35057
[13] Karch, G., Large-time behaviour of solutions to nonlinear wave equations: Higher-order asymptotics, Math. Methods Appl. Sci., 22, 18, 1671-1697 (1999) · Zbl 0947.35141
[14] Kobayashi, K.; Siaro, T.; Tanaka, H., On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29, 407-424 (1977) · Zbl 0353.35057
[15] Korpusov, M. O.; Sveshnikov, A. G., Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Zh. Vychisl. Mat. Mat. Fiz.. Zh. Vychisl. Mat. Mat. Fiz., Comput. Math. Math. Phys., 43, 12, 1765-1797 (2003), (in Russian); transl. in: · Zbl 1121.35329
[16] Korpusov, M. O.; Sveshnikov, A. G., Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Uravn.. Differ. Uravn., Differ. Equ., 42, 3, 431-443 (2006), (in Russian); transl. in: · Zbl 1131.35073
[17] Kwek, K. H.; Qu, C. C., Alternative principle for pseudo-parabolic equations, Dynam. Systems Appl., 5, 2, 211-217 (1996) · Zbl 0871.35054
[18] Levine, H. A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(P u_t = - A u + F(u)\), Arch. Ration. Mech. Anal., 51, 371-386 (1973) · Zbl 0278.35052
[19] Levine, H. A., The role of critical exponents in blowup theorems, SIAM Rev., 32, 2, 262-288 (1990) · Zbl 0706.35008
[20] Levine, H. A.; Zhang, Q. S., The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values, Proc. Roy. Soc. Edinburgh Sect. A, 130, 3, 591-602 (2000) · Zbl 0960.35051
[21] Martynenko, A. V.; Tedeev, A. F., The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density, Zh. Vychisl. Mat. Mat. Fiz.. Zh. Vychisl. Mat. Mat. Fiz., Comput. Math. Math. Phys., 47, 2, 238-248 (2007), (in Russian); transl. in: · Zbl 1210.35156
[22] Padron, V., Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356, 7, 2739-2756 (2004) · Zbl 1056.35103
[23] Ptashnyk, M., Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66, 12, 2653-2675 (2007) · Zbl 1112.35116
[24] Qi, Y. W., The critical exponents of parabolic equations and blow-up in \(R^n\), Proc. Roy. Soc. Edinburgh Sect. A, 128, 1, 123-136 (1998) · Zbl 0892.35088
[25] Quirós, F.; Rossi, J. D., Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions, Indiana Univ. Math. J., 50, 629-654 (2001) · Zbl 0994.35027
[26] Showalter, R. E.; Ting, T. W., Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1, 1-26 (1970) · Zbl 0199.42102
[27] Sobolev, S. L., On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18, 3-50 (1954) · Zbl 0055.08401
[28] Sviridyuk, G. A.; Fëdorov, V. E., Analytic semigroups with kernels, and linear equations of Sobolev type, Sibirsk. Mat. Zh., 36, 5, 1130-1145 (1995) · Zbl 0958.47019
[29] Ting, T. W., Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14, 1-26 (1963) · Zbl 0139.20105
[30] Ting, T. W., Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21, 440-453 (1969) · Zbl 0177.36701
[31] Wang, C. P.; Zheng, S. N., Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136, 2, 415-430 (2006) · Zbl 1104.35015
[32] Wang, C. P.; Zheng, S. N.; Wang, Z. J., Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20, 1343-1359 (2007) · Zbl 1173.35345
[33] Wang, Z. J.; Yin, J. X.; Wang, C. P.; Gao, H., Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources, J. Evol. Equ., 7, 615-648 (2007) · Zbl 1145.35041
[34] Winkler, M., A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25, 911-925 (2002) · Zbl 1007.35043
[35] Zheng, S. N.; Song, X. F.; Jiang, Z. X., Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298, 308-324 (2004) · Zbl 1078.35046
[36] Zheng, S. N.; Wang, C. P., Large time behavior of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21, 2179-2200 (2008) · Zbl 1149.35325
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.