Critical exponents for the heat conduction equation with a nonlinear boundary condition

We consider the heat conduction equation ut = ∇·(|∇u| m−1 ∇u )( m> 1),x∈ R N ,t >0, subject to the nonlinear boundary condition −|∇u| m−1 ∂u ∂x1 = u p ,x1 =0 ,t >0, in multi-dimension. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions. Mathematics Subject Classification: 35B33; 35K20; 35K57

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