Variational Approach to Complicated Similarity Solutions of Higher-Order Nonlinear PDEs. II

Some higher-order quasilinear parabolic, hyperbolic, and nonlinear dispersion equations are shown to admit various blow-up, extinction, and travelling wave solutions, which reduce to variational problems admitting countable families of compactly supported solutions.Comment: 48 pages, 26 figure

Three tupes of self-similar blow-up for the fourth-order p-Laplacian equaiton with source: variational and branching approaches

The fourth-order quasilinear reaction-diffusion equation with a p-Laplacian operator is shown to admit three types of blow-up. Self-similar patterns are first constructed for the regional blow-up case, where the rescaled problem admits a variational setting. Next, these were extended via a branching approach to non-variational problems.Comment: 39 pages, 24 figure

An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations

Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we pre- dict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two spatial dimensions, the singular perturbation theory reveals that the set of possible singularity points depends subtly on the geometry of the domain and the equation parameters. In three spatial dimensions, preliminary numerical simulations indicate that the multiplicity of singularities can be even more complex. For the aforementioned scenarios, the analysis highlights the dichotomy of behaviors exhibited between the second and fourth order cases.Comment: 32 Page

On the finite element method for a nonlocal degenerate parabolic problem

The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of degree $k\geq 1$. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment

Very singular solutions for the thin film equation with absorption

Self-similar large time behaviour of weak solutions of the fourth-order parabolic thin film equations with absorption is studued.Comment: 20 pages, 13 figure

Towards the KPP-Problem and log t-Front shift for Higher-Order Nonlinear PDEs II. Quasilinear Bi- and Tri-Harmonic Equations

It is shows that some aspects of classic KPP-problem (1937) can be extended to some fourth and sixth-order quasilinear parabolic equations.Comment: 26 pages, 15 figure

Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -\chi \nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = \Delta v - \mu + u \end{cases}\end{equation*} under no-flux boundary conditions in balls $\Omega\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overline{\Omega})$, where $\chi>0$ and $\mu:=\frac{1}{|\Omega|}\int_{\Omega}u_0$. Bellomo--Winkler (Comm.\ Partial Differential Equations;2017;42;436--473) proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when $p=q=1$ under some conditions for $\chi$ and $\int_\Omega u_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\geq 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$.Comment: 44 page

Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy

It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such extensions are not unique, this implies the principle nonuniqueness of shock-type solutions and also nonexistence of any entropy-type description of proper unique solutions. A difficult free-boundary setting, with extra conditions specified on shocks, are necessary to restore uniqueness in such problems.Comment: 19 pages, 8 figure

Towards the KPP--Problem and log t-Front Shift for Higher-Order Nonlinear PDEs III. Dispersion and Hyperbolic Equations

It is shown that some aspects of classic KPP-theory of 1937 can be applied to a number of higher-order dispersion, hyperbolic, and other equations.Comment: 25 pages, 19 figure

Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

We show that various asymptotic properties of global solutions of a fourth-order quasilinear thin film equation can be described by branching from corresponding solutions of the linear bi-harmonic equation. This includes a countable family of self-similar solutions and other aspects.Comment: 38 page