Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. (English) Zbl 1536.35101
Summary: We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation
\[
\partial_t u=\Delta u^m +|x|^{\sigma}u^p,
\]
posed for \((x,t)\in \mathbb{R}^N \times (0,T)\), with \(m>1, 1 \leq p<m\) and \(-2(p-1)/(m-1)<\sigma <\infty\). We prove that there are several types of self-similar solutions with respect to the local behavior near the origin, and their existence depends on the magnitude of \(\sigma\). In particular, these solutions have different blow-up sets and rates: some of them have \(x=0\) as a blow-up point, some other only blow up at (space) infinity. We thus emphasize on the effect of the weight on the specific form of the blow-up patterns of the equation. The present study generalizes previous works by the authors limited to dimension \(N=1\) and \(\sigma >0\).
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K65 | Degenerate parabolic equations |
References:
| [1] | D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa, 18 (1991), 363-441. · Zbl 0762.35052 |
| [2] | D. Andreucci and A. F. Tedeev, Universal bounds at the blow-up time for nonlinear parabolic equations, Adv. Differential Equations, 10 (2005), 89-120. · Zbl 1122.35042 |
| [3] | X. Bai, S. Zhou and S. Zheng, Cauchy problem for fast diffusion equation with localized reaction, Nonlinear Anal., 74 (2011), 2508-2514. · Zbl 1213.35074 |
| [4] | C. Bandle and H. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622. · Zbl 0693.35081 |
| [5] | P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. · Zbl 0556.35063 |
| [6] | P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. Differential Equations, 68 (1987), 238-252. · Zbl 0622.35033 |
| [7] | B. Ben Slimene, Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems, Differ. Equ. Appl., 11 (2019), 439-462. · Zbl 1435.35066 |
| [8] | B. Ben Slimene, S. Tayachi, and F. B. Weissler, Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations, Nonlinear Anal., 152 (2017), 116-148. · Zbl 1355.35020 |
| [9] | X. Cabré and Y. Martel, Existence versus explosion instantanée por des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris, 329 (1999), 973-978. · Zbl 0940.35105 |
| [10] | J. Carr, Applications of Centre Manifold Theory, Springer Verlag, New York, 1981. · Zbl 0464.58001 |
| [11] | N. Chikami, M. Ikeda, and K. Taniguchi, Well-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation, Nonlinearity, 34 (2021), 8094-8142. |
| [12] | N. Chikami, M. Ikeda, and K. Taniguchi, Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces, Nonlinear Anal., 222 (2022), paper no. 112931, 28pp. · Zbl 1491.35107 |
| [13] | C. Cortázar, M. del Pino, and M. Elgueta, On the blow-up set for ut = ∆u m + u m , m > 1, Indiana Univ. Math. J., 47 (1998), 541-562. · Zbl 0916.35056 |
| [14] | C. Cortázar, M. del Pino, and M. Elgueta, Uniqueness and stability of regional blow-up in a porous-medium equation, Ann. Inst. H. Poincaré Analyse Non Linéaire, 19 (2002), 927-960. · Zbl 1018.35062 |
| [15] | R. Ferreira and A. de Pablo, Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions, Rev. Mat. Complut., 31 (2018), 805-832. · Zbl 1400.35168 |
| [16] | R. Ferreira, A. de Pablo, and J. L. Vázquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Differential Equations, 231 (2006), 195-211. · Zbl 1110.35034 |
| [17] | S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonho-mogeneous nonlinearity, J. Differential Equations, 165 (2000), no. 2, 468-492. · Zbl 1011.34006 |
| [18] | Y. Giga and N. Umeda, Blow-up directions at space infinity for solutions of semilinear heat equations, Bol. Soc. Paran. Mat., 23 (2005), 9-28. · Zbl 1173.35531 |
| [19] | Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations, J. Math. Anal. Appl., 316 (2006), 538-555. · Zbl 1106.35029 |
| [20] | J. A. Goldstein and I. Kombe, Nonlinear degenerate prabolic equations with singular lower-order term, Adv. Differential Equations, 8 (2003), 1153-1192. · Zbl 1110.35036 |
| [21] | G. R. Goldstein, J. A. Goldstein, and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent, Appl. Anal., 84 (2005), 571-583. · Zbl 1071.35066 |
| [22] | J. Guckenheimer and Ph. Holmes, “Nonlinear Oscillation, Dynamical Systems and Bifurcations of Vector Fields,” Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. |
| [23] | J.-S. Guo, C.-S. Lin, and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient, Dynam. Systems Appl., 19 (2010), 415-433. · Zbl 1218.35047 |
| [24] | J.-S. Guo and M. Shimojo, Blowing up at zero points of potential for an initial bound-ary value problem, Commun. Pure Appl. Anal., 10 (2011), , 161-177. · Zbl 1229.35110 |
| [25] | J.-S. Guo, C.-S. Lin, and M. Shimojo, Blow-up for a reaction-diffusion equation with variable coefficient, Appl. Math. Lett., 26 (2013), 150-153. · Zbl 1254.35029 |
| [26] | J.-S. Guo, and P. Souplet, Excluding blowup at zero points of the potential by means of Liouville-type theorems, J. Differential Equations, 265 (2018), 4942-4964. · Zbl 1394.35229 |
| [27] | K. Hisa and J. Takahashi, Optimal singularities of initial data for solvability of the Hardy parabolic equation, J. Differential Equations, 296 (2021), 822-848. · Zbl 1469.35123 |
| [28] | R. G. Iagar and Ph. Laurençot, Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption, J. London Math. Soc., 87 (2013), 509-529. · Zbl 1264.35137 |
| [29] | R. G. Iagar, Ph. Laurençot, and A. Sánchez, Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with strong nonhomogeneous absorp-tion, Comm. Contemp. Math. (online ready), 2024. |
| [30] | R. G. Iagar, A. I. Muñoz, and A. Sánchez, Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential, J. Differential Equations, 358 (2023), 188-217. · Zbl 1511.35068 |
| [31] | R. G. Iagar, A. I. Muñoz, and A. Sánchez, Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension, Comm. Pure Appl. Analysis, 21 (2022), 891-925. · Zbl 1484.35108 |
| [32] | R. G. Iagar and A. Sánchez, Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction with linear growth, J. Dynam. Differential Equations, 31 (2019), 2061-2094. · Zbl 1430.35109 |
| [33] | R. G. Iagar and A. Sánchez, Instantaneous and finite time blow-up of solutions to a reaction-diffusion equation with Hardy-type singular potential, J. Math. Anal. Appl., 491 (2020), no. 1, paper no. 124244, 11 pages. · Zbl 1448.35053 |
| [34] | R. G. Iagar and A. Sánchez, Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction, J. Differential Equations, 272 (2021), 560-605. · Zbl 1454.35037 |
| [35] | R. G. Iagar and A. Sánchez, Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction, Nonlinear Anal., 217 (2022), paper no. 112740, 33 pages. · Zbl 1483.35047 |
| [36] | X. Kang, W. Wang, and X. Zhou, Classification of solutions of porous medium equa-tion with localized reaction in higher space dimensions, Differential Integral Equations, 24 (2011), 909-922. · Zbl 1249.35157 |
| [37] | I. Kombe, Doubly nonlinear parabolic equations with singular lower order term, Non-linear Anal., 56 (2004), 185-199. · Zbl 1047.35065 |
| [38] | A. A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Royal Society Edinburgh Sect. A, 98 (1984), 183-202. · Zbl 0556.35077 |
| [39] | H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equa-tions, J. Math. Kyoto Univ., 18 (1978), 221-227. · Zbl 0387.35008 |
| [40] | A. Mukai and Y. Seki, Refined construction of Type II blow-up solutions for semi-linear heat equations with Joseph-Lundgren supercritical nonlinearity, Discrete Cont. Dynamical Systems, 41 (2021), 4847-4885. · Zbl 1469.35063 |
| [41] | A. de Pablo and A. Sánchez, Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165 (2000), 377-413. · Zbl 0961.34011 |
| [42] | L. Perko, “Differential equations and dynamical systems,” Third edition, Texts in Applied Mathematics, 7, Springer Verlag, New York, 2001. · Zbl 0973.34001 |
| [43] | R. G. Pinsky, Existence and nonexistence of global solutions for ut = ∆u + a(x)u p in R d , J. Differential Equations, 133 (1997), 152-177. · Zbl 0876.35048 |
| [44] | R. G. Pinsky, The behavior of the life span for solutions to ut = ∆u + a(x)u p in R d , J. Differential Equations, 147 (1998), 30-57. · Zbl 0914.35056 |
| [45] | Y.-W. Qi, The critical exponents of parabolic equations and blow-up in R n , Proc. Royal Soc. Edinburgh A, 128 (1998), 123-136. · Zbl 0892.35088 |
| [46] | P. Quittner and Ph. Souplet, “Superlinear Parabolic Problems. Blow-up, Global Ex-istence and Steady States,” Birkhauser Advanced Texts, Birkhauser Verlag, Basel, 2007. · Zbl 1128.35003 |
| [47] | A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, “Blow-up in Quasilinear Parabolic Problems,” de Gruyter Expositions in Mathematics, 19, W. de Gruyter, Berlin, 1995. · Zbl 1020.35001 |
| [48] | J. Sotomayor, “Generic Bifurcations of Dynamical Systems,” in Proceedings of a Symposium Held at University of Bahia, Salvador, Brasil, Academic Press, New York, 1973, 561-582. · Zbl 0296.58007 |
| [49] | R. Suzuki, Existence and nonexistence of global solutions of quasilinear parabolic equa-tions, J. Math. Soc. Japan, 54 (2002), 747-792. · Zbl 1036.35085 |
| [50] | S. Tayachi, Uniqueness and non-uniqueness of solutions for critical Hardy-Hénon par-abolic equations, J. Math. Anal. Appl., 488 (2020), no. 1, paper no. 123976, 51 pages. · Zbl 1437.35437 |
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.
