Blow-up behaviour of one-dimensional semilinear parabolic equations

  • Herrero M
  • Velázquez J
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Abstract

Consider the Cauchy problem ut−uxx−F(u)=0;x∈ℝ,t>0u(x,0)=u0(x);x∈ℝwhere u0(x) is continuous, nonnegative and bounded, and F(u) = upwith p > 1, or F(u) = eu. Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) → (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(−x) for x > 0, there holds 1) If F(u) = upwith p > 1, then limt↑Tu(ξ((T−t)|log(T−t)|)1/2,t)×(T−t)1/(p−1)=(p−1)−(1/(p−1))[1+(p−1)ξ24p]−(1/(p−1))uniformly on compact sets |ξ| ≦ R with R > 0, 2) If F(u) = eu, then limt↑T(u(ξ((T−t)|log(T−t)|)1/2,t)+log(T−t))=−log[1+ξ24]uniformly on compact sets |ξ| ≦ R with R > 0.

Author-supplied keywords

  • Semilinear diffusion equations
  • asymptotic behaviour of solutions
  • blow-up

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