## New critical exponents for a doubly singular parabolic equation.(English)Zbl 1475.35188

Summary: This paper deals with the Cauchy problem for a doubly singular parabolic equation with nonlocal inner source $u_t = \mathrm{div}(|\nabla u^m|^{p-2} \nabla u^m) + \|u\|_{L^q(\mathbb{R}^N)}^{r-1} u^{s+1}, \quad (x,t) \in \mathbb{R}^N \times (0,T),$ where $$N \geq 1, 1-((1+m)/(1+mN)) < m(p-1) <1, 0<m \leq 1, q>1, r \geq 1, 0 \leq s < p/N + m(p-1) -1$$ and $$r + s>1$$. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.

### MSC:

 35K67 Singular parabolic equations 35B33 Critical exponents in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B44 Blow-up in context of PDEs 35K15 Initial value problems for second-order parabolic equations 35K59 Quasilinear parabolic equations 35K65 Degenerate parabolic equations 35R09 Integro-partial differential equations
Full Text:

### References:

 [1] Kalashnikov, AS., Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russ Math Surv, 42, 169-222 (1987) · Zbl 0642.35047 [2] Dibenedetto, E., Degenerate parabolic equations (1993), New York (NY): Springer, New York (NY) · Zbl 0794.35090 [3] Aronson, DG.The porous medium equation. In: Fasano A, Primicerio M, editors. Nonlinear diffusion problems. Berlin: Springer; 1986. · Zbl 0626.76097 [4] Pao, CV., Nonlinear parabolic and elliptic equations (1992), New York (NY): Plenum, New York (NY) [5] Furter, J.; Grinfield, M., Local vs non-local interactions in population dynamics, J Math Biol, 27, 65-80 (1989) · Zbl 0714.92012 [6] Fujita, H., On the blowing up of solution of the Cauchy problem for $$####$$, J Fac Sci Univ Tokyo Sect I, 13, 109-124 (1966) · Zbl 0163.34002 [7] Hayakawa, K., On nonexistence of global solutions of some semilinear parabolic equation, Proc Jpn Acad, 49, 503-505 (1973) · Zbl 0281.35039 [8] Weissler, FB., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J Math, 38, 29-40 (1981) · Zbl 0476.35043 [9] Galaktionov, VA; Kurdyumov, SP; Mikhailov, AP, Unbounded solutions of the Cauchy problem for the parabolic equation $$####$$, Soviet Phys Dokl, 25, 458-459 (1980) [10] Galaktionov, VA., Blow-up for quasilinear heat equations with critical Fujita’s exponent, Proc R Soc Edinburgh Sect A, 124, 3, 517-525 (1994) · Zbl 0808.35053 [11] Mochizuki, K.; Suzuki, R., Critical exponent and critical blow-up for quasilinear parabolic equations, Israel J Math, 98, 141-156 (1997) · Zbl 0880.35057 [12] Qi, YW., On the equation $$####$$, Proc Roy Soc Edinburgh Sect A, 123, 2, 373-390 (1993) · Zbl 0801.35068 [13] Mochizuki, K.; Mukai, K., Existence and nonexistence of global solutions to fast diffusions with source, Methods Appl Anal, 2, 92-102 (1995) · Zbl 0832.35083 [14] Galaktionov, VA., Conditions for nonexistence as a whole and localization of the solutions of Cauchy’s problem for a class of nonlinear parabolic equations, Zh Vychisl Mat Mat Fiz, 23, 1341-1354 (1985) [15] Qi, YW., Critical exponents of degenerate parabolic equations, Sci China Ser A, 10, 38, 1153-1162 (1995) · Zbl 0837.35076 [16] Qi, YW; Wang, MX., Critical exponents of quasilinear parabolic equations, J Math Anal Appl, 267, 264-280 (2002) · Zbl 1010.35005 [17] Galaktionov, VA; Levine, HA., A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal, 34, 1005-1027 (1998) · Zbl 1139.35317 [18] Liu, XF; Wang, MX., The critical exponent of doubly singular parabolic equations, J Math Anal Appl, 257, 170-188 (2001) · Zbl 0984.35020 [19] Afanas’eva, NV; Tedeev, AF., Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source, Ukr Math J, 57, 1687-1711 (2005) · Zbl 1094.35064 [20] Lee, TY; Ni, WM., Global existence, large time behavior and life span on solution of a semilinear parabolic Cauchy problem, Trans Am Math Soc, 333, 365-378 (1992) · Zbl 0785.35011 [21] Mu, CL; Li, YH; Wang, Y., Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values, Nonlinear Anal Real World Appl, 11, 198-206 (2010) · Zbl 1182.35028 [22] Yang, JG; Yang, CX; Zheng, SN., Second critical exponent for evolution p-Laplacian equation with weighted source, Math Comput Model, 56, 247-256 (2012) · Zbl 1255.35155 [23] Mukai, K.; Mochizuki, K.; Huang, Q., Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values, Nonlinear Anal, 39, 33-45 (2000) · Zbl 0936.35034 [24] Guo, JS; Guo, YJ., On a fast diffusion equation with source, Tohoku Math J, 53, 571-579 (2001) · Zbl 0995.35035 [25] Mu, CL; Zeng, R.; Zhou, SM., Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values, J Math Anal Appl, 384, 181-191 (2011) · Zbl 1227.35053 [26] Yang, CX; Ji, FY; Zhou, SS., The second critical exponent for a semilinear nonlocal parabolic equation, J Math Anal Appl, 418, 231-237 (2014) · Zbl 1310.35033 [27] Zhou, J., The second critical exponent for a nonlocal porous medium equation in $$####$$, Appl Math Lett, 38, 43-47 (2014) · Zbl 1328.35113 [28] Ma, LW; Fang, ZB., A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources, Commun Pure Appl Anal, 16, 1697-1706 (2017) · Zbl 1364.35167 [29] Ma, LW; Fang, ZB., Secondary critical exponent and life span for a nonlocal parabolic p-Laplace equation, Appl Anal, 97, 775-786 (2018) · Zbl 1391.35239
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.