Solution to nonlinear parabolic equations related to P-Laplacian

Abstract

Consider the following Cauchy problem: \begin{gathered} u_t = div(|\nabla u^m |^{p - 2} \nabla u^m ),(x,t) \in S_T = \mathbb{R}^N \times (0,T), \hfill \\ u(x,0) = \mu ,x \in \mathbb{R}^N \hfill \\ \end{gathered} where 1 < p < 2, 1 < m < \tfrac{1}{{p - 1}}, and µ is a σ-finite measure in ℝ ^N. By the Moser’s iteration method, the existence of the weak solution is obtained, provided that \tfrac{{(m + 1)N}}{{mN + 1}} < p. In contrast, if \tfrac{{(m + 1)N}}{{mN + 1}} \geqslant p, there is no solution to the Cauchy problem with an initial value δ(x), where δ(x) is the classical Dirac function.

Keywords

Nonlinear parabolic equation / Cauchy problem / Existence / σ-Finite measure

References

[1] Aronson D. G., Caffarelli L. A.. The initial trace of a solution of the porous medium equation. Trans. Amer. Soc., 1983, 280(1): 351-366

[2] Aronson D. G., Peletier L. A.. Large time behaviour of solutions of the porous medium equation in bounded domains. J. Differ. Equ., 1981, 39(3): 378-412

[3] Benilan Ph., Crandall M. G., Pierre M.. Solutions of the porous medium equation in ℝ^N under optimal conditions on initial values. Indiana Univ. Math. J., 1984, 33: 51-71

[4] Bernis F.. Existence results for doubly nonlinear higher order parabolic equations on unbounded domain. Math. Ann., 1988, 279: 373-394

[5] Chen Y.. Hölder continuity of the gradient of the solutions of certain degenerate parabolic equations. Chin. Ann. Math., 1987, 8B(3): 343-356

[6] Dahlberg B. E., Kenig C. E.. Nonnegative solutions of generalized porous medium equations. Rev. Math. Iberoamericana, 1986, 3: 267-305

[7] Dibenedetto E.. Degenerate Parabolic Equations, 1993, New York: Springer-Verlag

[8] Dibenedetto E., Friedman A.. Hölder estimates for nonlinear degenerate parabolic systems. J. Reine. Angew. Math., 1985, 357: 1-22

[9] Dibenedetto E., Herrero M. A.. On Cauchy problem and initial traces for a degenerate parabolic equations. Trans. Amer. Soc., 1989, 314: 187-224

[10] Dibenedetto E., Herrero M. A.. Nonnegative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when 1 < p < 2. Rational Mech. Anal., 1990, 111(3): 225-290

[11] Fan H.. Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure. Acta Math. Sin. (Engl. Ser.), 2004, 20: 663-682

[12] Filo J.. Local existence and L _∞-estimate of weak solutions to a nonlinear degenerate parabolic equation with nonlinear boundary data. Panamer Math. J., 1994, 4: 1-31

[13] Giaquinta M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, 1983, Princeton: Princeton Univ. Press.

[14] Gmira A.. On quasilinear parabolic equations involving measure data. Asymptot. Anal., 1990, 3: 43-56

[15] Herrero M. A., Pierre M.. The Cauchy problem for u _t = Δu ^m when 0 < m < 1. Trans. Amer. Math. Soc., 1985, 291: 145-158

[16] Ishige K.. On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation. SIAM J. Math. Anal., 1996, 27: 1235-1260

[17] Ivanov A. V.. Hölder continuity of solutions for nonlinear degenerate parabolic equations. J. Sovit. Math., 1991, 56(2): 2320-2347

[18] Ladyzenskaja O. A., Solonnikov V. A., Ural’ceva N. N.. Linear and Quasilinear Equations of Parabolic Type, 1968, Providence, R. I.: Amer. Math. Soc.

[19] Li Y., Xie Ch.. Blow-up for p-Laplace parabolic equations. Electron. J. Differential Equations, 2003, 2003(20): 1-12

[20] Vazquez J. L.. Smoothing and decay estimates for nonlinear diffusion equations, 2006, Oxford: Oxford University Press

[21] Vazquez J. L.. The Porous Medium Equation, Oxford Math. Monographs, 2007, Oxford: Clarendon Press

[22] Wu Z., Zhao J., Yun J., Li F.. Nonlinear Diffusion Equations, 2001, New York, Singapore: World Scientific Publishing

[23] Yan J., Zhao J.. A note to the evolutional P-Laplace equation with absorption (in Chinese). Acta. Sci. Nat. Jilin, 1995, 33(2): 35-38

[24] Yuan H.. Hölder continuity of solutions for nonlinear degenerate parabolic equations (in Chinese). Acta. Sci. Nat. Jilin, 1991, 29(2): 36-52

[25] Yuan H., Lian S., Cao C. Extinction and positivity for a doubly nonlinear degenerate parabolic equation. Acta Math. Sin. (Engl. Ser.), 2007, 23: 1751-1756

[26] Zhan H.. Harnack estimates for weak solutions to a singular parabolic equation. Chin. Ann. Math., 2011, 32B(3): 397-416

[27] Zhao J.. Existence and nonexistence of solution for u ^t = div(|∇u| ^p−2∇u)+f(∇u, u, x, t). J. Math. Anal. Appl., 1993, 172: 130-146

[28] Zhao J.. Source-type solutions of quasilinear degenerate parabolic equation with absorption. Chin. Ann. Math., 1994, 15B(1): 89-104

[29] Zhao J.. The Cauchy problem for u _t = div(|∇u| ^p−2∇u) when \tfrac{{2N}}{{N + 1}} < p < 2. Nonlinear Anal., 1995, 24: 615-630

[30] Zhao J., Xu Z.. Cauchy problem and initial traces for a doubly degenerate parabolic equation. Sci. China Ser. A, 1996, 39: 673-684

[31] Zhao J., Yuan H.. The Cauchy problem of a class of doubly degenerate parabolic equation (in Chinese). Chin. Ann. Math., 1995, 16A(2): 181-196