Existence of solutions to some degenerate parabolic equation associated with the \(p\)-Laplacian in the critical case. (English) Zbl 1286.35143
Summary: This paper is concerned with the initial-boundary value problem for the following degenerate parabolic equation: \(u_t(x,t)-\varDelta_pu(x,t)-| u|^{q-2}u(x,t)=f(x,t)\) with initial data \(u_0\in L^r(\varOmega )\). G. Akagi [J. Differ. Equations 241, No. 2, 359–385 (2007; Zbl 1145.35074)] established the existence of local (in time) solutions to this problem in the case \(r>N(q-p)/p\); however, the critical case \(r=N(q-p)/p\) has been left as an open problem. In this paper, even in the critical case \(r=N(q-p)/p\), the existence of solutions to the problem is established under a certain restriction on \(u_0\). The key to our proof is Tartar’s inequality, which enables us to derive desired convergences of approximate solutions to the problem from the compactness of the embedding \(W_0^{1,p}(\varOmega )\subset L^2(\varOmega )\). Incidentally, any smoothness is not imposed on \(\partial\varOmega \) at all while a smooth boundary is needed in [loc. cit.].
MSC:
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35K65 | Degenerate parabolic equations |
| 34G25 | Evolution inclusions |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Citations:
Zbl 1145.35074References:
| [1] | Akagi, G., Local existence of solutions to some degenerate parabolic equation associated with the \(p\)-Laplacian, J. Differential Equations, 241, 359-385 (2007) · Zbl 1145.35074 |
| [2] | Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \Delta u + u^{1 + \alpha}\), J. Fac. Sci. Univ. Tokyo Sec. I, 13, 109-124 (1966) · Zbl 0163.34002 |
| [3] | Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci., 8, 211-229 (1972) · Zbl 0248.35074 |
| [4] | Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034 |
| [5] | Weissler, F. B., Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32, 277-296 (1979) · Zbl 0419.47031 |
| [6] | Brézis, H.; Cazenave, T., A nonlinear heat equation with singular initial data, J. Anal. Math., 68, 277-304 (1996) · Zbl 0868.35058 |
| [7] | Ôtani, M., On existence of strong solutions for \(d u(t) / d t + \partial \psi^1(u(t)) - \partial \psi^2(u(t)) \ni f(t)\), J. Fac. Sci. Univ. Tokyo Sec. 1A, 24, 575-605 (1977) · Zbl 0386.47040 |
| [8] | Ni, W. M.; Sacks, P., Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc., 287, 657-671 (1985) · Zbl 0573.35046 |
| [9] | Brézis, H., Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam · Zbl 0252.47055 |
| [10] | Brézis, H.; Crandall, M. G.; Pazy, A., Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 23, 123-144 (1970) · Zbl 0182.47501 |
| [11] | Akagi, G.; Ôtani, M., Evolution inclusions governed by subdifferentials in reflexive Banach spaces, J. Evol. Equ., 4, 519-541 (2004) · Zbl 1081.34059 |
| [12] | Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 |
| [13] | Okazawa, N.; Yokota, T., Global existence and smoothing effect for the complex Ginzburg-Landau equation with \(p\)-Laplacian, J. Differential Equations, 182, 541-576 (2002) · Zbl 1005.35086 |
| [14] | Sakaguchi, S., Coincidence sets in the obstacle problem for the \(p\)-harmonic operator, Proc. Amer. Math. Soc., 95, 382-386 (1985) |
| [15] | Simon, J., Régularité de la solution d’une équation non liniéaire dans \(R^N\), (Journées d’ Analyse Non Liniéaire (Besancon, 1977). Journées d’ Analyse Non Liniéaire (Besancon, 1977), Lecture Notes in Mathematics, vol. 665 (1978), Springer-Verlag: Springer-Verlag Berlin and New York), 205-227 · Zbl 0402.35017 |
| [16] | Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51, 126-150 (1984) · Zbl 0488.35017 |
| [17] | Kenmochi, N., Some nonlinear parabolic variational inequalities, Israel J. Math., 22, 304-331 (1975) · Zbl 0327.49004 |
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