Self-similar solutions for the heat equation with a positive non-Lipschitz continuous, semilinear source term

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We investigate the existence of self-similar solutions for the parabolic equation u t = Δ u + u m H u , with 0 ≤ m < 1 and H the Heaviside graph, coupled with the initial datum u x , 0 = − c x 2 1 1 − m , with c > 0 . We analyze two cases: the problem in R n , n > 1 , with m = 0 and the problem in R when 0 ≤ m < 1 . In the first case we extend the result of Gianni and Hulshof (1992) and show that there exist only two self-similar solutions changing sign, provided 0 < c < c c r , with c c r obtained solving a specific algebraic equation depending on n . In the second case we prove that there exist at least two self-similar solutions of problem u t = u x x + u m H u , u x , 0 = − c x 2 1 1 − m , changing sign and evolving region where u > 0 . These solutions are of great interest. Indeed, on one hand they prove that the problem does not admit uniqueness and on the other they prove that a single point where u x , 0 = 0 , for an initial datum which is otherwise negative, can generate a region where u x , t is positive.

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