## The blow-up rate for a system of heat equations with nonlinear boundary conditions.(English)Zbl 0941.35008

The authors establish the blow-up rate and locate the blow-up set of some positive solutions of the system $$u_t=\Delta u$$, $$v_t=\Delta v$$ for $$x\in B_R=\{y:|y|<R\}$$, $$t>0$$, complemented by the boundary conditions $$\partial u/\partial\nu=v^p$$, $$\partial v/\partial\nu=u^q$$. Here $$p,q>0$$, $$pq>1$$, and the initial conditions $$u_0,v_0$$ are assumed to be radially symmetric, positive, to satisfy the compatibility conditions and $$\Delta u_0,\Delta v_0\geq 0$$. The blow-up rate of $$(\max u(\cdot,t),\max v(\cdot,t))$$ is of order $$((T-t)^{-\alpha/2},(T-t)^{-\beta/2})$$, where $$T$$ is the blow-up time, $$\alpha=(p+1)/(pq-1)$$, $$\beta=(q+1)/(pq-1)$$ and blow-up occurs only at the boundary. Similar results for $$p,q\geq 1$$ were obtained in [K. Deng, Z. Angew. Math. Phys. 47, No. 1, 132-143 (1996; Zbl 0854.35054)].

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K50 Systems of parabolic equations, boundary value problems (MSC2000)

Zbl 0854.35054
Full Text:

### References:

 [1] Levine, H.A; Payne, L.E, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. differential equations, 16, 319-334, (1974) · Zbl 0285.35035 [2] Levine, H.A; Smith, R.A, A potential well theory for the heat equation with a nonlinear boundary condition, Math. methods appl. sci., 9, 127-136, (1987) · Zbl 0646.35049 [3] Walter, W, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. math. anal., 6, 85-90, (1975) · Zbl 0268.35052 [4] Gomez, J.L; Marquez, V; Wolanski, N, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. differential equation, 92, 384-401, (1991) · Zbl 0735.35016 [5] Fila, M; Quittner, P, The blow-up rate for heat equation with a nonlinear boundary condition, Math. methods appl. sci., 14, 197-205, (1991) · Zbl 0735.35014 [6] Hu, B; Yin, H.M, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Transactions amer. soc., 346, 117-135, (1994) · Zbl 0823.35020 [7] Deng, K, Global existence and blow-up for a system of heat equations with nonlinear boundary conditions, Math. methods appl. sci., 18, 307-315, (1995) · Zbl 0822.35074 [8] Giga, Y; Kohn, R.V, Asympotic self-similar blow-up of semilinear heat equations, Comm. pure appl. math., 38, 297-319, (1985) · Zbl 0585.35051 [9] Pao, C.V, Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York [10] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. · Zbl 0144.34903
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