Self-Similar Blow-up in Higher-Order Semilinear Parabolic Equations

• Published in: SIAM journal on applied mathematics Vol. 64; no. 5; pp. 1775 - 1809
• Main Authors: Budd, C. J., Galaktionov, V. A., Williams, J. F.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2004
• Subjects: Adjoints; Applied mathematics; Coefficients; Eigenfunctions; Eigenvalues; Explosions; Mathematics; Nonlinearity; Odes; Ordinary differential equations; Phase transitions; Polynomials; Solvability
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/S003613990241552X

We study the Cauchy problem in R × R for one-dimensional 2mth-order, m > 1, semilinear parabolic PDEs of the form ($D_{x} = \partial/\partial x) u_t = (-1)^{m+1} D_x^{2m}u + \mid u\mid^{p-1}u$, where p > 1, and$u_{t} = (-1)^{m+1}D_{x}^{2m}u + e^u$with bounded initial data$u_{0}(x)$. Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical second-order parabolic equations$u_{t} = u_{xx} + u^p$and$u_{t} = u_{xx} + e^u$from combustion theory, the blow-up in their higher-order counterparts is asymptotically self-similar. In particular, there exist exact nontrivial self-similar blow-up solutions,$u_{*}(x,t) = (T-t)^{-1/(p-1)}) f(y)$in the case of the polynomial nonlinearity and u(x, t) = -ln(T - t) + f(y) for the exponential nonlinearity, where$y = x/(T - t)^{1/2m}$is the backward higher-order heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non-self-adjoint higher-order linear differential operators. We show that there are at least$2[\frac{m}{2}]$nontrivial self-similar solutions to the full PDEs. Numerical solution of the ODEs for m = 2 and 3 supports this, and the time dependent solutions of the PDEs for m = 2 are then studied by using a scale invariant adaptive numerical method. It is shown that those functions f(y), which have the simplest spatial shape (e.g., a single maximum), correspond to stable self-similar solutions. A further countable subset of nonsimilarity blow-up patterns can be constructed by linearization and matching with similarity solutions of a first-order Hamilton-Jacobi equation.