Abstract
We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.
Highlights
This is joint work with my student Ryuichi Sato (Tohoku University) and it is concerned with the heat equation with a nonlinear boundary condition
[1]–[5], [7]–[11], [12]–[17], [18], [19], [20] and references therein) while there are few results related to the dependence of the blow-up time on the initial function even in the case Ω = RN+
We remark that the blow-up time for problem (1) cannot be chosen uniform for all initial functions lying in a bounded set of Lr(RN+ ) with 1 ≤ r ≤ N (p − 1)
Summary
This is joint work with my student Ryuichi Sato (Tohoku University) and it is concerned with the heat equation with a nonlinear boundary condition,. For any φ ∈ BU C(Ω), problem (1) has a unique solution u ∈ C2,1(Ω × (0, T ]) ∩ C1,0(Ω × (0, T ]) ∩ BU C(Ω × [0, T ]) For some T > 0 and the maximal existence time T (φ) of the solution can be defined. T→ T (φ) and we call T (φ) the blow-up time of the solution u.
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