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

In this paper, we study the similar entropy solutions of the singular diffusion equation, @?u@?t=@?@?x@j(@?u@?x), with @j(s)=s/1+s^2. These kinds of solutions have nonvertical jump lines. We establish the existence and uniqueness and also discuss some properties of these kinds of solutions.

Dry gel assisting crystallization of bifunctional CuO–ZnO–Al2O3/SiO2–Al2O3 catalysts for CO2 hydrogenation

This paper considers the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term, which is a product of localized source, local source, weight function, and complemented by homogeneous Dirichlet boundary conditions. It is proved that there exists initial data such that simultaneous or non-simultaneous blow-up occur. Moreover, the related classifications for the four parameters in the model is optimal and complete. The results extend those in Zhang and Yang (J. Appl. Math. Comput. 32:429–441, 2010).

Simultaneous and non-simultaneous blow-up criteria of solutions for a diffusion system with weighted localized sources

This paper is concerned with the existence of positive periodic solutions to
a nonlinear fourth-order differential equation. By virtue of the first
positive eigenvalue of the linear equation corresponding to the nonlinear
fourth order equation, we establish the existence result by using the fixed
point index theory in a cone.

Zejia Wang

Papers

Similar entropy solutions of a singular diffusion equation

Dry gel assisting crystallization of bifunctional CuO–ZnO–Al2O3/SiO2–Al2O3 catalysts for CO2 hydrogenation

Simultaneous and non-simultaneous blow-up criteria of solutions for a diffusion system with weighted localized sources

Positive periodic solutions to a nonlinear fourth-order differential equation

The simultaneous and non-simultaneous blow-up criteria for a diffusion system