Helsingin Yliopisto


Long-time asymptotics of linear and semilinear diffusion equations

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Department of Mathematics and Statistics of University of Helsinki


Formula 1. A classical semilinear Cauchy problem.


The new result on the linear case is based on abstract functional analytic methods. The study of the long-time asymptotic behaviour of semilinear parabolic Cauchy-problems is based on explicit integration techniques together with weighted sup-norm estimates, applied to the estimation of nonlinear terms in the conventional integral equation (obtained form Duhamel's principle).


  • J.Bonet, W.Lusky, J.Taskinen:
    Schauder basis and decay rate of the heat equation.

    We consider the (very!) classical linear heat or diffusion equation with integrable Cauchy data g in the Eulidean space R^n. For example if the space dimension n is one, the explicit solution formula implies the large time t decay rate t^{-1/2}, and it is known that if for example the integral of g over the real line vanishes, then the decay rate is faster, t^{-1}. Vanishing higher iterated integrals imply faster decay rates. In this paper we consider initial data in weighted L^p(R^n)-spaces, where 1 < p < \infty and the weight is fast growing at the infinity. Our result says that given arbirary natural number N, the space can be decomposed to two components X and Y such that X is finite dimensional and if the initial data belongs to Y, then the corresponding solution of the heat equation decays at least at the rate t^{-N}.
    The above mentioned decomposition can be done by perturbing any Schauder basis of the initial data space. The case of p=1 is also considered, with a bit different methods.
    Here is a preprint.

  • Long time asymptotics of sub-threshold solutions of a semilinear Cauchy problem.
    Diff.Eq.Appl. 3,2 (2011), 279-297.

    Restricting to the case of positive initial data (and thus solution), we relax the assumption on the smallness of initial data in the paper below. All sub-threshold data h can be treated at least if h is an even function. Moreover, in addition to p ≥ 4 we also consider the cases 3 < p < 4. Here, the perturbation term of the solution is shown to be of the order t^(2-p)/2 instead of 1/t.
    Recently, a related paper was published by Ishige, Ishiwata and Kawakami, Indiana Univ. Math. J. (2009).

  • Aymptotical behaviour of a class of semilinear diffusion equations.
    J.Evol.Equations 7,3 (2007), 429-447.

    We show that for small initial data, the unique solution of the semilinear Cauchy-problem as in Formula 1 behaves like the Gaussian solution of the linear heat equation (which is of order 1/√t for t → ∞ in space dimension 1) plus a perturbation, which is of order 1/t only.
    The result is obtained as follows. Using the Duhamel principle we write the standard integral equation corresponding to the Cauchy problem. Using tricky integrations by parts, several terms are obtained. It turns out possible to process and sort them such that those with order 1/√t behaviour are exactly of Gaussian heat solution type, and the rest of the terms are only of order 1/t -type.
    The estimation techniques for the perturbative terms emerge from the Cahn-Hilliard paper (2005). We use weighted sup-norms for these estimates.
    We restrict the considerations to one space dimension due to inessential technical complications. However, the method applies to very general types of nonlinearities (examples are given in the paper) as long as the nonlinearity is of polynomial nature and of polynomial order at least 4.
    Besides higher dimensions, the adoptation of this "explicit" method to more general parabolic equations remains as an interesting challenge.

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