# American Institute of Mathematical Sciences

August  2021, 14(8): 2607-2623. doi: 10.3934/dcdss.2021032

## Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term

 1 Université de Sfax, Faculté des Sciences de Sfax, Département de Mathematiques, BP 1171, Sfax 3000, Tunisia 2 Université Sorbonne Paris Nord, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, 99 avenue J.B. Clément, 93430 Villetaneuse, France

Dedicated to the memory of Ezzeddine Zahrouni

Received  September 2020 Revised  February 2021 Published  August 2021 Early access  March 2021

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work [1], we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

Citation: Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032
##### References:
 [1] B. Abdelhedi and H. Zaag, Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term, J. Differential Equations, 272 (2021), 1-45.  doi: 10.1016/j.jde.2020.09.020. [2] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser.(2), 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473. [3] M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.  doi: 10.1002/cpa.3160410606. [4] J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011. [5] M. Chipot and F. B. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060. [6] G. K. Duong and H. Zaag, Profile of touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci., 29 (2019), 1279-1348.  doi: 10.1142/S0218202519500222. [7] S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of $u_t-\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.  doi: 10.1002/cpa.3160450703. [8] S. Filippas and W. X. Liu., On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 313-344.  doi: 10.1016/S0294-1449(16)30215-3. [9] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. [10] V. A. Galaktionov and J. L. Vázquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.  doi: 10.1137/0524071. [11] V. A. Galaktionov and J. L. Vázquez, Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations, 127 (1996), 1-40.  doi: 10.1006/jdeq.1996.0059. [12] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.  doi: 10.1002/cpa.3160380304. [13] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.  doi: 10.1512/iumj.1987.36.36001. [14] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.  doi: 10.1002/cpa.3160420607. [15] M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997. [16] M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450. [17] M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 131-189.  doi: 10.1016/S0294-1449(16)30217-7. [18] M. A. Herrero and J. J. L. Velázquez, Comportement générique au voisinage d'un point d'explosion pour des solutions d'équations paraboliques unidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 201-203. [19] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.  doi: 10.1002/cpa.3160450303. [20] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t = \Delta u +|u|^{p-1} u$, Duke Math. J., 86 (1997), 143-195. [21] F. Merle and H. Zaag, Stabilité du profil à l'explosion pour les équations du type $u_ t = \Delta u+\vert u\vert ^ {p-1}u$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 345-350. [22] V. T. Nguyen, Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time, Phys. D, 339 (2017), 49-65.  doi: 10.1016/j.physd.2016.09.002. [23] V. T. Nguyen and H. Zaag, Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method, Anal. PDE, 9 (2016), 229-257.  doi: 10.2140/apde.2016.9.229. [24] P. Quittner and P. Souplet, Superlinear parabolic Problems. Blow-up, Global Existence and Steady States, Second Edition. Birkhäuser Advanced Texts, 2019. doi: 10.1007/978-3-030-18222-9. [25] P. Souplet, S. Tayachi and F. B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45 (1996), 655-682.  doi: 10.1512/iumj.1996.45.1197. [26] S. Tayachi and H. Zaag, Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Trans. Amer. Math. Soc., 371 (2019), 5899-5972.  doi: 10.1090/tran/7631. [27] S. Tayachi and H. Zaag, Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term, Actes du Colloque EDP-Normandie, Le Havre, 21–22, octobre 2015. [28] F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224.  doi: 10.1016/0022-0396(84)90081-0. [29] H. Zaag, Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 581-622.  doi: 10.1016/S0294-1449(98)80002-4.

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##### References:
 [1] B. Abdelhedi and H. Zaag, Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term, J. Differential Equations, 272 (2021), 1-45.  doi: 10.1016/j.jde.2020.09.020. [2] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser.(2), 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473. [3] M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.  doi: 10.1002/cpa.3160410606. [4] J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011. [5] M. Chipot and F. B. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060. [6] G. K. Duong and H. Zaag, Profile of touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci., 29 (2019), 1279-1348.  doi: 10.1142/S0218202519500222. [7] S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of $u_t-\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.  doi: 10.1002/cpa.3160450703. [8] S. Filippas and W. X. Liu., On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 313-344.  doi: 10.1016/S0294-1449(16)30215-3. [9] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. [10] V. A. Galaktionov and J. L. Vázquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24 (1993), 1254-1276.  doi: 10.1137/0524071. [11] V. A. Galaktionov and J. L. Vázquez, Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations, 127 (1996), 1-40.  doi: 10.1006/jdeq.1996.0059. [12] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.  doi: 10.1002/cpa.3160380304. [13] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.  doi: 10.1512/iumj.1987.36.36001. [14] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.  doi: 10.1002/cpa.3160420607. [15] M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973-997. [16] M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381-450. [17] M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 131-189.  doi: 10.1016/S0294-1449(16)30217-7. [18] M. A. Herrero and J. J. L. Velázquez, Comportement générique au voisinage d'un point d'explosion pour des solutions d'équations paraboliques unidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 201-203. [19] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math., 45 (1992), 263-300.  doi: 10.1002/cpa.3160450303. [20] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t = \Delta u +|u|^{p-1} u$, Duke Math. J., 86 (1997), 143-195. [21] F. Merle and H. Zaag, Stabilité du profil à l'explosion pour les équations du type $u_ t = \Delta u+\vert u\vert ^ {p-1}u$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 345-350. [22] V. T. Nguyen, Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time, Phys. D, 339 (2017), 49-65.  doi: 10.1016/j.physd.2016.09.002. [23] V. T. Nguyen and H. Zaag, Blow-up results for a strongly perturbed semilinear heat equation: Theoretical analysis and numerical method, Anal. PDE, 9 (2016), 229-257.  doi: 10.2140/apde.2016.9.229. [24] P. Quittner and P. Souplet, Superlinear parabolic Problems. Blow-up, Global Existence and Steady States, Second Edition. Birkhäuser Advanced Texts, 2019. doi: 10.1007/978-3-030-18222-9. [25] P. Souplet, S. Tayachi and F. B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45 (1996), 655-682.  doi: 10.1512/iumj.1996.45.1197. [26] S. Tayachi and H. Zaag, Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Trans. Amer. Math. Soc., 371 (2019), 5899-5972.  doi: 10.1090/tran/7631. [27] S. Tayachi and H. Zaag, Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term, Actes du Colloque EDP-Normandie, Le Havre, 21–22, octobre 2015. [28] F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224.  doi: 10.1016/0022-0396(84)90081-0. [29] H. Zaag, Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 581-622.  doi: 10.1016/S0294-1449(98)80002-4.
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