Publication: Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions.
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Taylor & Francis
In this work we analyze the existence of solutions that blow-up in finite time for a reaction-diffusion equation ut−Δu=f(x,u) in a smooth domain Ω with nonlinear boundary conditions ∂u∂n=g(x,u). We show that, if locally around some point of the boundary, we have f(x,u)=−βup,β≥0, and g(x,u)=uq, then blow-up in finite time occurs if 2q>p+1 or if 2q=p+1 and β<q. Moreover, if we denote by Tb the blow-up time, we show that a proper continuation of the blow-up solutions are pinned to the value infinity for some time interval [T,τ] with Tb≤T<τ. On the other hand, for the case f(x,u)=−βup, for all x and u, with β>0 and p>1, we show that blow-up occurs only on the boundary.
Andreu, F., Mazon, J. M., Toledo, J., Rossi, J. D. (2002). Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal. TMA 49:541–563. Arrieta, J. M., Carvalho, A., Rodríguez-Bernal, A. (2000). Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds. Comm. Partial Differential Equations 25(1/2):1–37. Baras, P., Cohen, L. (1987). Complete blow-up after Tmax for the solution of a semilinear heat equation. J. Funct. Anal. 71(1):142–174. Brezis, H. Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: Zarantonello, E., ed. Contributions to Nonlinear Functional Analysis, pp. 101–165. Chipot, M., Fila, M., Quittner, P. (1991). Stationary solution, Blow-up and convergence to stationary solution for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. U. Comenian. 60:35–103. Courant, Hilbert, D. (1953). Methods of Mathematical Physics. Vol. I. New York: Wiley-Interscience. Evans, L. (1977). Regularity properties of the heat equation subject to nonlinear boundary constraints.Nonlinear Anal. Theory Methods Appl. 1:593–602. Friedman, A., McLeod, B. (1985). Blow-up of positive solutions of semilinear heat equations. Indiana U. Math. J. 34(2):425–447. Galaktionov, V., Vazquez, J. L. (1997). Continuation of blow up solutions of nonlinear heat equations in several space dimensions. Comm. Pure Appl. Math 50:1–67. Galaktionov, V., Vazquez, J. L. (2002). The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 8(2):399–433. García-Melián, J., Gómez-Re~nasco, R., López-Gómez, J., Sabina de Lis, J. (1998). Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs.Arch. Ration. Mech. Anal. 145(3):261–289. García-Melián, J., Letelier-Albornoz, R., Sabina de Lis, J. (2001). Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up. Proc. Am. Math. Soc. 129(12):3593–3602. Keller, J. B. (1957). On solutions of Δu=f(u). Comm. Pure Appl. Math. 10:503–510. Osserman, R. (1957). On the inequality Δu≥f(u). Pacific J. Math. 7:1641–1647. Quiros, F., Rossi, J. D., Vazquez, J. L. (2002). Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions. Comm. Partial Differential Equations 27:395–424. Rodríguez-Bernal, A. (2002). Attractors parabolic problems with nonlinear boundary conditions critical exponents and singular initial data. J. Differential Equations 181:165–196. Rodríguez-Bernal, A., Tajdine, A. (2001). Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differential Equations 169:332–372. Veron, L. (1992). Semilinear elliptic equations with uniform blow-up on the boundary. J. Anal. Math. 59:231–250.