On Asymptotic “Eigenfunctions” of the Cauchy Problem for a Nonlinear Parabolic Equation

Stability and dynamics of self-similarity in evolution equations

“…As seen in (3.3), absorption scales the amplitude of the solution proportional to t −1/q while diffusion does so proportional to t −1/2 . The mechanism that reduces the temperature most rapidly will dominate (see the discussions in [38,50,19,20,72,41]). Specifically: − For q > 2 diffusion dominates and the solution approaches a Gaussian (3.4) whose amplitude decays like t −1/2 as t → ∞.…”

“…These boundary conditions are motivated by the exponentially localized initial conditions considered earlier. If one relaxes these conditions, there exist a one-parameter family of selfsimilar solutions with algebraic decay in the far-field, some with finite mass and some with infinite mass [19,20,38]; these solutions will not be discussed here. Self-similar solutions are s-independent steady states of this problem, U =Ū (η),…”

“…We will concentrate on (1.2) where g(u) = −|u| q u is an absorption term. For this class of equations, global existence is easily established and one can prove the existence of a unique attracting exponentially-localized infinite-time self-similar spreading solution for (1.2) and its generalization to higher dimensions [38,50,19]. Different ranges for values of the parameter q yield either diffusion-or absorption-dominated dynamics.…”

Large Time Behavior of Small Solutions to Dirichlet Problem for Landau-Ginzburg Type Equations

“…Blow-up in finite time of positive solutions to the Cauchy problem for the heat equation u t À Du ¼ u 1þs was proved in [3], [8], [15], [22]. Large time behavior of positive solutions for nonlinear heat equation (which is a particular case of (2)) u t À Du þ u 1þs ¼ 0 was studied for any s > 0 (see [13] for the super critical case s > 2=n, [4], [7], [14] for the critical case s ¼ 2=n and [1], [2] for the sub critical case s A ð0; 2=nÞ). Global existence in time of small solutions to (2) in the super critical case s > 2=n was also shown in [3].…”

“…More precisely, if ðarg a; arg bÞ ¼ ð0; 0Þ, ðb > 0Þ, then these problems have solutions globally in time (see [4] for (2) and [17] for (1) with n ¼ 1) and if ðarg a; arg bÞ ¼ ð0;GpÞ, ðb < 0Þ, then solutions of problems (1) and (2) blow up in finite time (see [3], [8], [15] for (2) and [18], [19], [20] for (1), for review of blow up results, see [16]). Another situation we have in the case of complex coe‰cients a; b, since there are points ðarg a; arg bÞ which satisfy (4) but do not (6).…”