Opuscula Math. 39, no. 3 (2019), 395-414
https://doi.org/10.7494/OpMath.2019.39.3.395
Opuscula Mathematica
Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
Abstract. We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.
Keywords: degenerate quasilinear parabolic equation, nonlinear source term, Moser's method.
Mathematics Subject Classification: 35B40, 35D35, 58J35, 58K30.
- N.D. Alikakos, \(L^p\)-bound of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827-868.
- N.D. Alikakos, R. Rostamian, Gradient estimates for degenerate diffusion equations, Math. Ann. 259 (1982), 827-868.
- D. Andreucci, A.F. Tedeev, A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary, J. Math. Anal. Appl. 231 (1999), 543-567.
- E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York, 1993.
- Z. Junning, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differential Equations 102 (1993), 35-52.
- H. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 37 (1990), 262-288.
- G.M. Lieberman, Time-periodic solutions of quasilinear parabolic differential equations, J. Math. Anal. Appl. 264 (2001), 617-638.
- S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, New York, 1973.
- M. Nakao, Global solutions for some nonlinear parabolic equations with non-monotonic perturbations, Nonlinear Anal. 10 (1986), 455-466.
- M. Nakao, Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term, J. Math. Soc. Japan 64 (2012), 851-883.
- M. Nakao, Existence of global decaying solutions to the exterior problem for the Klein-Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity, J. Differential Equations 255 (2013), 3940-3970.
- M. Nakao, Global existence to the initial-boundary value problem for a system of semilinear wave equations, Nonlinear Analysis TMA 146 (2016), 233-257.
- M. Nakao, On the initial-boundary value problem for some quasilinear parabolic equations of divergence form, J. Differential Equations 263 (2017), 8565-8580.
- M. Nakao, Global existence to the initial-boundary value problem for a system of nonlinear diffusion and wave equations, J. Differential Equations 264 (2018), 134-162.
- M. Nakao, Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations, J. Math. Anal. Appl. 462 (2018), 1585-1604.
- M. Nakao, C. Chen, Global existence and gradient estimates for the quasilinear parabolic equations of \(m\)-Laplacian type with a nonlinear convection term, J. Differential Equations 162 (2000), 224-250.
- M. Nakao, A. Naimah, On global attractor for nonlinear parabolic equations of \(m\)-Laplacian type, J. Math. Anal. Appl. 331 (2007), 793-809.
- M. Nakao, Y. Ohara, Gradient estimates of periodic solutions for some quasilinear parabolic equations, J. Math. Anal. Appl. 204 (1996), 868-883.
- Y. Ohara, \(L^{\infty}\) estimates of solutions of some nonlinear degenerate parabolic equations, Nonlinear Anal. 18 (1992), 413-426.
- M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations 46 (1982), 268-299.
- M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RIMS, Kyoto Univ. 8 (1972), 211-229.
- M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988), 187-212.
- L. Véron, Coercivité et propriétés régularisantes des semi-groupes non-linéaires dans les espaces de Banach, Faculte des Sciences et Techniques, Université Francois Rabelais, Tours, France, 1976.
- Mitsuhiro Nakao
- Kyushu University, Faculty of Mathematics, Moto-oka 744, Fukuoka 819-0395, Japan
- Communicated by Marius Ghergu.
- Received: 2018-05-28.
- Accepted: 2018-08-14.
- Published online: 2019-02-23.
