Resumen de Existence and Multiplicity of Blow-Up Profiles for a Quasilinear Diffusion Equation with Source

Razvan-Gabriel Iagar, Ariel Sánchez

• We classify radially symmetric self-similar solutions presenting finite time blow-up to the quasilinear diffusion equation with weighted source ut = um + |x| σ u p, posed for (x, t) ∈ RN × (0, T ), T > 0, in dimension N ≥ 1 and in the range of exponents −2 <σ < ∞, 1 < m < p < ps(σ ), where ps(σ ) = m(N+2σ+2) N−2 , N ≥ 3, +∞, N ∈ {1, 2}, is the renowned Sobolev critical exponent. The most interesting result is the multiplicity of two different types of self-similar solutions for p sufficiently close to m and σ sufficiently close to zero in dimension N ≥ 2, including solutions with dead-core profiles. For σ = 0, this answers in dimension N ≥ 2 a question still left open in Samarskii et al. (Blow-up in quasilinear parabolic problems. de Gruyter expositions in mathematics, W. de Gruyter, Berlin, 1995, Section IV.1.4, pp. 195–196), where only multiplicity in dimension N = 1 had been established. Besides this result, we also prove that, for any σ ∈ (−2, 0), N ≥ 1 and m < p < ps(σ ) existence of at least a self-similar blow-up solution is granted. In strong contrast with the previous results, given any N ≥ 1, σ ≥ σ∗ = (mN + 2)/(m − 1) and p ∈ (m, ps(σ )), non-existence of any radially symmetric self-similar solution is proved.