TY - JOUR

T1 - Heat equation with a nonlinear boundary condition and growing initial data

AU - Ishige, Kazuhiro

AU - Sato, Ryuichi

N1 - Funding Information:
for all sufficiently large κ. Thus Corollary 1.3 follows. □ Acknowledgements. The authors would like to thank the referees for their valuable comments. The first author was supported in part by the Grant-in-Aid for Scientific Research (A)(No. 15H02058), from Japan Society for the Promotion of Science. The second author was supported in part by Research Fellow of Japan Society for the Promotion of Science.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition ∂tu = Δ x ∈ Ω t > 0; ∇u .v(x) = up; x ∈ ∂ Δ t > 0;(x; 0) ∝ = (x) x ∈ Ω where N ≥ 1, p > 1, is a smooth domain in RN and '(x) = O(eλd(x)2 ) as d(x) → 1 for some λ ≥ 0. Here, d(x) = dist (x; ∈ Ω). Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary ∈ Ω.

AB - We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition ∂tu = Δ x ∈ Ω t > 0; ∇u .v(x) = up; x ∈ ∂ Δ t > 0;(x; 0) ∝ = (x) x ∈ Ω where N ≥ 1, p > 1, is a smooth domain in RN and '(x) = O(eλd(x)2 ) as d(x) → 1 for some λ ≥ 0. Here, d(x) = dist (x; ∈ Ω). Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary ∈ Ω.

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M3 - Article

AN - SCOPUS:85028703481

VL - 30

SP - 481

EP - 504

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 7-8

ER -