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Asymptotic behaviour of solutions of some semilinear parabolic problems. (English) Zbl 0918.35025

Summary: We consider the Cauchy problem: \[ u_t-\Delta u+ u^p= 0\quad\text{for }x\in \mathbb{R}^N,\quad t>0,\tag{1} \]
\[ u(x,0)= u_0(x)\quad\text{for }x\in \mathbb{R}^N.\tag{2} \] Here \(p>1\), \(N\geq 1\) and \(u_0(x)\) is a continuous, nonnegative and bounded function such that: \[ u_0(x)\sim A| x|^{-\alpha},\quad\text{as }| x|\to \infty,\tag{3} \] for some \(A>0\) and \(\alpha>0\). In this paper, we discuss the asymptotic behaviour of solutions to (1)–(3) in terms of the various values of the parameters \(p\), \(N\), \(\alpha\) and \(A\). A common pattern that emerges from our analysis is the existence of an external zone where \(u(x,t)\sim u_0(x)\) and one (or several) internal regions, where the influence of diffusion and absorption is most strongly felt. We present a complete classification of the size of these regions, as well as that of the stabilization profiles that unfold therein, in terms of the aforementioned parameters.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
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References:

[1] Abramowitz, M.; Stegun, I.A., ()
[2] Brezis, H., Semilinear equations in \(\textit{R}N\) without conditions at infinity, Appl. math. optim, Vol. 12, 271-282, (1984) · Zbl 0562.35035
[3] Brezis, H.; Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, Journal de mathématiques pures et appliquées, Vol. 62, 73-97, (1983) · Zbl 0527.35043
[4] Bender, C.M.; Orszag, S.A., Advanced mathematical methods for scientists and engineers, (1978), McGraw-Hill Inc · Zbl 0417.34001
[5] Brezis, H.; Peletier, L.A.; Terman, D., A very singular solution of the heat equation with absorption, Arch. rat. mech. anal, Vol. 95, 185-209, (1986) · Zbl 0627.35046
[6] Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear analysis T. M. A, Vol. 11, 1103-1133, (1987) · Zbl 0639.35038
[7] Escobedo, M.; Kavian, O.; Matano, H., Large time behavior of solutions of a dissipative semilinear heat equation, Comm. in PDE, Vol. 20, 1427-1452, (1995) · Zbl 0838.35015
[8] Friedman, A., Partial differential equations of parabolic type, (1983), Robert E. Krieger Pub. Co
[9] Giga, Y.; Kohn, R.V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. pure appl. math, Vol. 38, 287-319, (1985) · Zbl 0585.35051
[10] Galaktionov, V.A.; Kurdyumov, S.P.; Samarskii, A.A., On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation, Math. USSR sbornik, Vol. 54, 421-455, (1986) · Zbl 0607.35049
[11] Galindo, A.; Pascual, P., Quantum mechanics, (1991), Springer Verlag · Zbl 0824.00009
[12] Gmira, A.; Veron, L., Large time behaviour of the solutions of a semilinear parabolic equation in \(\textit{R}N\), Journal of differential equations, Vol. 53, 258-276, (1984) · Zbl 0529.35041
[13] Herrero, M.A.; Velazquez, J.J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. inst. Henri Poincaré, Vol. 10, 2, 131-189, (1993) · Zbl 0813.35007
[14] Kamin, S.; Peletier, L.A., Large time behaviour of solutions of the heat equation with absorption, Annali scuola normale sup. di Pisa, Vol. XII, 3, 393-408, (1985) · Zbl 0598.35050
[15] Kamin, S.; Ughi, M., On the behaviour as t → ∞ of the solutions of the Cauchy problem for certain nonlinear parabolic equations, Journal of mathematical analysis and applications, Vol. 128, 456-469, (1987) · Zbl 0643.35048
[16] Velazquez, J.J.L., Classification of singularities for blowing up solutions in higher dimensions, Trans. of the A. M. S, Vol. 338, 1, 441-464, (1993) · Zbl 0803.35015
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