Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 1 (2016), 229-257.

Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method

Van Tien Nguyen and Hatem Zaag

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precise blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn. Note that our method is applicable to more general equations, in particular those with no scaling invariance.

Article information

Source
Anal. PDE, Volume 9, Number 1 (2016), 229-257.

Dates
Received: 25 November 2014
Revised: 20 August 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843208

Digital Object Identifier
doi:10.2140/apde.2016.9.229

Mathematical Reviews number (MathSciNet)
MR3461306

Zentralblatt MATH identifier
1334.35148

Subjects
Primary: 35K10: Second-order parabolic equations
Secondary: 35K58: Semilinear parabolic equations

Keywords
blow-up Lyapunov functional asymptotic behavior blow-up profile semilinear heat equation lower-order term

Citation

Nguyen, Van Tien; Zaag, Hatem. Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method. Anal. PDE 9 (2016), no. 1, 229--257. doi:10.2140/apde.2016.9.229. https://projecteuclid.org/euclid.apde/1510843208


Export citation

References

  • L. M. Abia, J. C. López-Marcos, and J. Martínez, “On the blow-up time convergence of semidiscretizations of reaction–diffusion equations”, Appl. Numer. Math. 26:4 (1998), 399–414.
  • L. M. Abia, J. C. López-Marcos, and J. Martínez, “The Euler method in the numerical integration of reaction–diffusion problems with blow-up”, Appl. Numer. Math. 38:3 (2001), 287–313.
  • J. M. Ball, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations”, Quart. J. Math. Oxford Ser. $(2)$ 28:112 (1977), 473–486.
  • G. Baruch, G. Fibich, and N. Gavish, “Singular standing-ring solutions of nonlinear partial differential equations”, Phys. D 239:20-22 (2010), 1968–1983.
  • J. Bebernes and S. Bricher, “Final time blowup profiles for semilinear parabolic equations via center manifold theory”, SIAM J. Math. Anal. 23:4 (1992), 852–869.
  • M. Berger and R. V. Kohn, “A rescaling algorithm for the numerical calculation of blowing-up solutions”, Comm. Pure Appl. Math. 41:6 (1988), 841–863.
  • A. Bressan, “On the asymptotic shape of blow-up”, Indiana Univ. Math. J. 39:4 (1990), 947–960.
  • A. Bressan, “Stable blow-up patterns”, J. Differential Equ. 98:1 (1992), 57–75.
  • J. Bricmont and A. Kupiainen, “Universality in blow-up for nonlinear heat equations”, Nonlinearity 7:2 (1994), 539–575.
  • A. Cangiani, E. H. Georgoulis, I. Kyza, and S. Metcalfe, “Adaptivity and blow-up detection for nonlinear non-stationary convection-diffusion problems”, in preparation.
  • C. Fermanian Kammerer and H. Zaag, “Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation”, Nonlinearity 13:4 (2000), 1189–1216.
  • C. Fermanian Kammerer, F. Merle, and H. Zaag, “Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view”, Math. Ann. 317:2 (2000), 347–387.
  • S. Filippas and R. V. Kohn, “Refined asymptotics for the blowup of $u\sb t-\Delta u=u\sp p$”, Comm. Pure Appl. Math. 45:7 (1992), 821–869.
  • S. Filippas and W. X. Liu, “On the blowup of multidimensional semilinear heat equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 10:3 (1993), 313–344.
  • H. Fujita, “On the blowing up of solutions of the Cauchy problem for $u\sb{t}=\Delta u+u\sp{1+\alpha }$”, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.
  • Y. Giga and R. V. Kohn, “Characterizing blowup using similarity variables”, Indiana Univ. Math. J. 36:1 (1987), 1–40.
  • Y. Giga and R. V. Kohn, “Nondegeneracy of blowup for semilinear heat equations”, Comm. Pure Appl. Math. 42:6 (1989), 845–884.
  • Y. Giga, S. Matsui, and S. Sasayama, “Blow up rate for semilinear heat equations with subcritical nonlinearity”, Indiana Univ. Math. J. 53:2 (2004), 483–514.
  • Y. Giga, S. Matsui, and S. Sasayama, “On blow-up rate for sign-changing solutions in a convex domain”, Math. Methods Appl. Sci. 27:15 (2004), 1771–1782.
  • P. Groisman, “Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions”, Computing 76:3–4 (2006), 325–352.
  • P. Groisman and J. D. Rossi, “Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions”, J. Comput. Appl. Math. 135:1 (2001), 135–155.
  • P. Groisman and J. D. Rossi, “Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem”, Asymptot. Anal. 37:1 (2004), 79–91.
  • M. A. Hamza and H. Zaag, “Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case”, J. Hyperbolic Differ. Equ. 9:2 (2012), 195–221.
  • M. A. Hamza and H. Zaag, “A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations”, Nonlinearity 25:9 (2012), 2759–2773.
  • M. A. Herrero and J. J. L. Velázquez, “Blow-up profiles in one-dimensional, semilinear parabolic problems”, Comm. Partial Differential Equations 17:1–2 (1992), 205–219.
  • 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. 314:3 (1992), 201–203.
  • M. A. Herrero and J. J. L. Velázquez, “Flat blow-up in one-dimensional semilinear heat equations”, Differential Integral Equations 5:5 (1992), 973–997.
  • M. A. Herrero and J. J. L. Velázquez, “Generic behaviour of one-dimensional blow up patterns”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 19:3 (1992), 381–450.
  • M. A. Herrero and J. J. L. Velázquez, “Blow-up behaviour of one-dimensional semilinear parabolic equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 10:2 (1993), 131–189.
  • I. Kyza and C. Makridakis, “Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations”, SIAM J. Numer. Anal. 49:1 (2011), 405–426.
  • H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu\sb{t}=-Au+{\mathscr F}(u)$”, Arch. Rational Mech. Anal. 51 (1973), 371–386.
  • N. Masmoudi and H. Zaag, “Blow-up profile for the complex Ginzburg–Landau equation”, J. Funct. Anal. 255:7 (2008), 1613–1666.
  • F. Merle and H. Zaag, “Stability of the blow-up profile for equations of the type $u\sb t=\Delta u+\vert u\vert \sp {p-1}u$”, Duke Math. J. 86:1 (1997), 143–195.
  • F. K. N'gohisse and T. K. Boni, “Numerical blow-up for a nonlinear heat equation”, Acta Math. Sin. $($Engl. Ser.$)$ 27:5 (2011), 845–862.
  • V. T. Nguyen, “Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time”, preprint, 2014.
  • V. T. Nguyen, “On the blow-up results for a class of strongly perturbed semilinear heat equations”, Discrete Contin. Dyn. Syst. 35:8 (2015), 3585–3626.
  • V. T. Nguyen and H. Zaag, “Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations”, preprint, 2014. To appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci.
  • J. J. L. Velázquez, “Higher-dimensional blow up for semilinear parabolic equations”, Comm. Partial Differential Equations 17:9–10 (1992), 1567–1596.
  • J. J. L. Velázquez, “Classification of singularities for blowing up solutions in higher dimensions”, Trans. Amer. Math. Soc. 338:1 (1993), 441–464.
  • F. B. Weissler, “Existence and nonexistence of global solutions for a semilinear heat equation”, Israel J. Math. 38:1–2 (1981), 29–40.