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

Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition

In this paper, we establish the blow-up theorems of the Fujita type for a class of homogeneous Neumann problems of quasilinear equations with convection terms. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited such that the critical Fujita exponent even could be infinite for the model considered in the paper owing to the effect of convection.

http://iopscience.iop.org/article/10.1088/0951-7715/20/6/002/pdf

Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold.

Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

This paper is concerned with the p-Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.

Positive radial solutions of p-Laplacian equation with sign changing nonlinear sources

This paper is concerned with the porous medium equations for coupling via nonlinear boundary flux; we obtain the critical global existence curve and the critical Fujita curve for the problem, considered by constructing the self-similar supersolutions and subsolutions.

Zejia Wang

Papers

Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition

Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data

Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

Positive radial solutions of p-Laplacian equation with sign changing nonlinear sources

Critical exponents for porous medium systems coupled via nonlinear boundary flux