## Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term.(English)Zbl 1423.35186

This article studies the equation $u_t = \Delta u + |u|^{p-1} u + \mu |\nabla u|^q$ with $$\mu>0$$ in $$\mathbb{R}^N\times(0,T)$$, which in some sense is an intermediate between semilinear heat and Hamilton-Jacobi equations. More precisely, the article is concerned with the critical case $$q=\frac{2p}{p+1}$$ for $$p>3$$.
Here, a new blow-up profile (different from those occurring in the case of $$\mu=0$$) appears. In this article it is first motivated by a formal approach, afterwards the existence of a solution approaching this profile is proven: If initial data are suitably chosen, the difference between solution and profile is ‘trapped’ in a set shrinking to zero. (The choice of this trap, along with parabolic estimates adapted to its geometry, is one of the novelties of the paper, but also causes the restriction on $$p$$, whose necessity remains unknown.)
The method also gives stability of this solution.
Proofs are presented such that first an outline is given and technical results are provided in more detail later.

### MSC:

 35K55 Nonlinear parabolic equations 35B44 Blow-up in context of PDEs
Full Text:

### References:

 [1] Abramowitz, Milton; Stegun, Irene A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, xiv+1046 pp. (1964), For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. · Zbl 0171.38503 [2] Alfonsi, Liliane; Weissler, Fred B., Blow up in $${\bf R}^n$$ for a parabolic equation with a damping nonlinear gradient term. Nonlinear diffusion equations and their equilibrium states, 3, Gregynog, 1989, Progr. Nonlinear Differential Equations Appl. 7, 1-20 (1992), Birkh\"auser Boston, Boston, MA · Zbl 0795.35051 [3] Berger, Marsha; Kohn, Robert V., A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41, 6, 841-863 (1988) · Zbl 0652.65070 [4] Bricmont, J.; Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity, 7, 2, 539-575 (1994) · Zbl 0857.35018 [5] Chipot, M.; Weissler, F. B., Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20, 4, 886-907 (1989) · Zbl 0682.35010 [6] Chleb\'\i k, Miroslav; Fila, Marek; Quittner, Pavol, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, 4, 525-537 (2003) · Zbl 1028.35071 [7] C\^ote, Rapha\"el; Zaag, Hatem, Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., 66, 10, 1541-1581 (2013) · Zbl 1295.35124 [8] Ebde, M. A.; Zaag, H., Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term, SeMA J., 55, 5-21 (2011) · Zbl 1241.35125 [9] Fila, Marek, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc., 111, 3, 795-801 (1991) · Zbl 0768.35047 [10] Galaktionov, V. A.; Posashkov, S. A., The equation $$u_t=u_{xx}+u^\beta$$. Localization, asymptotic behavior of unbounded solutions, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint, 97, 30 pp pp. (1985) [11] Galaktionov, V. A.; Posashkov, S. A., Application of new comparison theorems to the investigation of unbounded solutions of nonlinear parabolic equations, Differentsial\cprime nye Uravneniya, 22, 7, 1165-1173, 1285; English transl., Differential Equation 22 (1986), 809-815 (1986) [12] Galaktionov, Victor A.; V\'azquez, Juan L., Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24, 5, 1254-1276 (1993) · Zbl 0813.35033 [13] Galaktionov, Victor A.; Vazquez, Juan L., Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations, 127, 1, 1-40 (1996) · Zbl 0884.35014 [14] Giga, Yoshikazu; Kohn, Robert V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 3, 297-319 (1985) · Zbl 0585.35051 [15] Giga, Yoshikazu; Kohn, Robert V., Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36, 1, 1-40 (1987) · Zbl 0601.35052 [16] Giga, Yoshikazu; Kohn, Robert V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42, 6, 845-884 (1989) · Zbl 0703.35020 [17] Herrero, M. A.; Vel\'azquez, J. J. L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 10, 2, 131-189 (1993) · Zbl 0813.35007 [18] Kawohl, B.; Peletier, L. A., Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z., 202, 2, 207-217 (1989) · Zbl 0661.35053 [19] Marcus, Moshe; Nguyen, Phuoc-Tai, Elliptic equations with nonlinear absorption depending on the solution and its gradient, Proc. Lond. Math. Soc. (3), 111, 1, 205-239 (2015) · Zbl 1326.35124 [20] Masmoudi, Nader; Zaag, Hatem, Blow-up profile for the complex Ginzburg-Landau equation, J. Funct. Anal., 255, 7, 1613-1666 (2008) · Zbl 1158.35016 [21] Merle, Frank, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45, 3, 263-300 (1992) · Zbl 0785.35012 [22] Merle, Frank; Rapha\“el, Pierre; Rodnianski, Igor, Blow up dynamics for smooth equivariant solutions to the energy critical Schr\'”odinger map, C. R. Math. Acad. Sci. Paris, 349, 5-6, 279-283 (2011) · Zbl 1213.35139 [23] Merle, Frank; Zaag, Hatem, Stability of the blow-up profile for equations of the type $$u_t=\Delta u+|u|^{p-1}u$$, Duke Math. J., 86, 1, 143-195 (1997) · Zbl 0872.35049 [24] NguyenZV. T. Nguyen and H. Zaag, Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations, arXiv:1406.5233, (2014). submitted. [25] Nouaili, Nejla; Zaag, Hatem, Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation, Comm. Partial Differential Equations, 40, 7, 1197-1217 (2015) · Zbl 1335.35126 [26] Rapha\“el, Pierre; Rodnianski, Igor, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes \'”Etudes Sci., 115, 1-122 (2012) · Zbl 1284.35358 [27] Rapha\"el, Pierre; Schweyer, Remi, Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Comm. Pure Appl. Math., 66, 3, 414-480 (2013) · Zbl 1270.35136 [28] Rapha\`“el, Pierre; Schweyer, R\'”emi, On the stability of critical chemotactic aggregation, Math. Ann., 359, 1-2, 267-377 (2014) · Zbl 1320.35100 [29] Serrin, James; Zou, Henghui, Existence and nonexistence results for ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal., 121, 2, 101-130 (1992) · Zbl 0795.35027 [30] Simon, Barry, Functional integration and quantum physics, Pure and Applied Mathematics 86, ix+296 pp. (1979), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0434.28013 [31] Snoussi, Seifeddine; Tayachi, Slim, Large time behavior of solutions for parabolic equations with nonlinear gradient terms, Hokkaido Math. J., 36, 2, 311-344 (2007) · Zbl 1180.35113 [32] Snoussi, S.; Tayachi, S.; Weissler, F. B., Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term, Proc. Roy. Soc. Edinburgh Sect. A, 129, 6, 1291-1307 (1999) · Zbl 0939.35089 [33] Souplet, Philippe, Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19, 16, 1317-1333 (1996) · Zbl 0858.35067 [34] Souplet, Philippe, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, No. 10, 19 pp pp. (2001) · Zbl 0982.35054 [35] Souplet, Philippe, The influence of gradient perturbations on blow-up asymptotics in semilinear parabolic problems: a survey. Nonlinear elliptic and parabolic problems, Progr. Nonlinear Differential Equations Appl. 64, 473-495 (2005), Birkh\"auser, Basel · Zbl 1094.35060 [36] Souplet, Philippe; Tayachi, Slim, Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88, 1, 135-154 (2001) · Zbl 0984.35077 [37] Souplet, Philippe; Tayachi, Slim; Weissler, Fred B., Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45, 3, 655-682 (1996) · Zbl 0990.35061 [38] Souplet, Philippe; Weissler, Fred B., Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212, 1, 60-74 (1997) · Zbl 0892.35011 [39] Souplet, Philippe; Weissler, Fred B., Poincar\'e’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 16, 3, 335-371 (1999) · Zbl 0924.35065 [40] Vel\'azquez, J. J. L., Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., 338, 1, 441-464 (1993) · Zbl 0803.35015 [41] Weissler, Fred B., Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55, 2, 204-224 (1984) · Zbl 0555.35061 [42] Nguyen, V. T., Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time, Phys. D, 339, 49-65 (2017) · Zbl 1376.35088 [43] Zaag, Hatem, Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 15, 5, 581-622 (1998) · Zbl 0902.35050
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.