Rossi, Julio D. The blow-up rate for a system of heat equations with non-trivial coupling at the boundary. (English) Zbl 0870.35050 Math. Methods Appl. Sci. 20, No. 1, 1-11 (1997). Let \(P=(p_{ij})\) be a matrix of nonnegative entries such that \(p_{11}<1\), \(p_{22}<1\) and \(\text{det}(P-Id)<0\). Let \((\alpha_1,\alpha_2)\) be a vector of negative entries such that \((P-Id)\) \((\alpha_1,\alpha_2)^T= (-1,-1)^T\). The paper establishes that any positive solution of the problem \(\partial_tv_i= \partial^2_rv_i+ (N-1)r^{-1}\partial_rv_i\) with \(\partial_rv_i(0,t)= 0\), \(\partial_rv_i(1,t)= v^{p_{i1}}_1(1,t)v^{p_{i2}}_2(1,t)\) and \(v_i(r,0)= v_{i0}(r)\) for \(i\in\{1,2\}\), \(r\in[0,1]\), \(t\in[0,T[\), and \(v_{i0}\in C^3\), such that \(\partial_rv_i\), \(\partial_tv_i\), \(\partial_t\partial_rv_i\geq 0\), satisfies \(c\leq v_i(1,t)(T-t)^{-\alpha_i/2}\leq C\) as \(t\to T\). Reviewer: N.A.Watson (Christchurch) Cited in 1 ReviewCited in 20 Documents MSC: 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs PDFBibTeX XMLCite \textit{J. D. Rossi}, Math. Methods Appl. Sci. 20, No. 1, 1--11 (1997; Zbl 0870.35050) Full Text: DOI References: [1] and , Blow-up vs. global existence for quasilinear parabolic systems with a nonlinear boundary condition, preprint (1995). [2] Amann, Math. Z 202 pp 219– (1989) · Zbl 0702.35125 · doi:10.1007/BF01215256 [3] Deng, Math. Meth. in the Appl. Sci. 18 pp 307– (1995) · Zbl 0822.35074 · doi:10.1002/mma.1670180405 [4] Deng, Z. Angew Math. Phys. 46 pp 110– (1995) · Zbl 0840.58013 · doi:10.1007/BF00917874 [5] Deng, Acta Math. Univ Comenianae LXIII pp 169– (1994) [6] Fila, Math. Meth. in the Appl. Sci. 14 pp 197– (1991) · Zbl 0735.35014 · doi:10.1002/mma.1670140304 [7] Hu, Trans. Amer. Math. Soc. 346 pp 117– (1995) · doi:10.1090/S0002-9947-1994-1270664-3 [8] Ladyzenskaja, Trans. Math. Monographs 23 (1968) [9] Levine, Diff. Eq. 16 pp 319– (1974) · Zbl 0285.35035 · doi:10.1016/0022-0396(74)90018-7 [10] Lopez, J. Diff. Eq. 92 pp 384– (1991) · Zbl 0735.35016 · doi:10.1016/0022-0396(91)90056-F [11] and , ’Blow-up results and localization of blow-up points in an N-dimensional smooth domain’, Preprint (1995). [12] and , ’Localization of blow-up points for a parabolic system with nonlinear boundary condition’, Preprint (1996). [13] ’On existence and nonexistence in the large, for an N-dimensional system of heat equations with nontrivial coupling at the boundary’, Preprint (1995). [14] Rossi, Diff. Int. Eq. [15] Walter, SIAM J. Math. Anal. 6 pp 85– (1975) · Zbl 0268.35052 · doi:10.1137/0506008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.