Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Instability of the anomalies in the one-dimensional Anderson model at weak disorder

We study the asymptotic behavior of the invariant measure, the Lyapunov exponent, and the density of states in the weak disorder limit in the case where the single-site potential distribution μ is not centered and for the special energies E =cos( πp/q ). We also prove that in general the above quantities can be continuously extended to zero disorder as continuous functions in the disorder parameter for all energies E ∈(−1, 1).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45159/1/10955_2005_Article_BF01029199.pd