Cauchy problems of semilinear pseudo-parabolic equations ✩

: This report summarizes mathematical research in the physical and biological sciences. Topics include: Partial differential equations; Parabolic and elliptic equations; Diffusion processes; Convective systems; Nonlinear waves; Free boundary problems.

Nonlinear Diffusion Equations.

A miscellany of new strategies, experimental techniques and theoretical approaches are emerging in the ongoing battle against cancer. Nevertheless, as new, ground-breaking discoveries relating to many and diverse areas of cancer research are made, scientists often have recourse to mathematical modelling in order to elucidate and interpret these experimental findings. Indeed, experimentalists and clinicians alike are becoming increasingly aware of the possibilities afforded by mathematical modelling, recognising that current medical techniques and experimental approaches are often unable to distinguish between various possible mechanisms underlying important aspects of tumour development. 
This short treatise presents a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti-cancer treatments. 
Thus, a selection of mathematical publications, including the influential theoretical studies by Burton, Greenspan, Liotta et al., McElwain and co-workers, Adam and Maggelakis, and Byrne and co-workers are juxtaposed with the seminal experimental findings of Gray et al. on oxygenation and radio-sensitivity, Folkman on angiogenesis, Dorie et al. on cell migration and a wide variety of other crucial discoveries. In this way the development of this field of research through the interactions of these different approaches is illuminated, demonstrating the origins of our current understanding of the disease.

A history of the study of solid tumour growth : the contribution of mathematical modelling

In this paper, we investigate the large time behaviour of solutions to the exterior problems of a class of quasilinear parabolic equations with convection terms. We establish the critical Fujita exponents pc and blow-up theorems of the Fujita type for both homogeneous Neumann and Dirichlet problems. In particular, it is shown that the critical p = pc belongs to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited that the critical Fujita exponent pc could even be infinite for the considered model because of the nonlinear convection.

https://iopscience.iop.org/article/10.1088/0951-7715/21/9/015/pdf

Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms

This paper deals with the critical blow-up and extinction ex- ponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents q1,q2 2 (0,+1) with q1 < q2. In other words, when q belongs to dierent intervals (0,q1),(q1,q2),(q2,+1), the solution possesses complete dierent prop- erties. More precisely speaking, as far as the blow-up exponent is con- cerned, the global existence case consists of the interval (0,q2). However, when q 2 (q2,+1), there exist both global solutions and blow-up so- lutions. As for the extinction exponent, the extinction case happens to the interval (q1,+1), while for q 2 (0,q1), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = q1 is concerned, the other parameter ‚ will play an im- portant role. In other words, when ‚ belongs to dierent interval (0 ,‚1) or (‚1,+1), where ‚1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely dierent properties.

/pdf/critical-blow-up-and-extinction-exponents-for-non-newton-2ol1uja88v.pdf

Critical blow-up and extinction exponents for non-newton polytropic filtration equation with source

Existence of weak solutions for a class of nonlinear diffusion problems

This paper deals with a degenerate parabolic system with nonlocal sources. It is prove that if pc = p1q1 − (m − p2)(n − q2) 0, there are solutions that blow-up and others that are global. When pc = 0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

Blow-up and Global Existence of Solutions for a Nonlocal Degenerate Parabolic System

MULTIPLICITY OF SOLUTIONS OF WEIGHTED (p, q)-LAPLACIAN WITH SMALL SOURCE

This paper deals with the large time behavior of solutions to the fast diffusive Newtonian filtration equations coupled via the nonlinear boundary sources. The result of Fujita type is obtained by constructing various kinds of upper and lower solutions. Particularly, we show that the critical global existence curve and
the critical Fujita curve are the same for the multi-dimensional system, which is quite different from the known results for corresponding one-dimensional problem.

10.1017/S000497271200038X

/pdf/critical-curves-for-a-coupled-system-of-fast-diffusive-4cx7t1ypol.pdf

Zejia Wang

Papers

Existence of weak solutions for a class of nonlinear diffusion problems

Blow-up and Global Existence of Solutions for a Nonlocal Degenerate Parabolic System

MULTIPLICITY OF SOLUTIONS OF WEIGHTED (p, q)-LAPLACIAN WITH SMALL SOURCE

Critical curves for a coupled system of fast diffusive newtonian filtration equations

Symmetry breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation