Source-type Solutions of the Heat Equation with Nonlinear Convection in n-space Dimensions (original) (raw)
In this paper we study the existence or nonexistence of a source-type solution for the heat equation with nonlinear convection: ut = ∆u+~b ·∇u , (x, t) ∈ ST = IR N × (0, T ], u(x, 0) = δ(x), x ∈ IR , where δ(x) denotes Dirac-delta measure in IR , N ≥ 2, n ≥ 0 and ~b = (b1, · · · , bN) ∈ IR N is a constant vector. It is shown that there exists a critical number pc = N+2 N such that the source-type solution to the above problem exists and is unique if 0 ≤ n < pc, while such solution does not exist if n ≥ pc. Moreover, the asymptotic behavior of the solution near origin, when it exists, is derived. It is shown that the source-type solution near the origin has the same behavior as that for the heat equation without convection if 0 ≤ n < N+1 N .
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