We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
In this paper, we consider the following PDE-ODE system modelling cancer invasion with slow diffusion and ECM remodelling,\[ \begin{cases} u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), \\ v_t=\Delta v+u-v, \\ \omega_t={-}v\omega+\eta \omega(1-u-\omega). \end{cases} \]For the special case $\eta =0$
, fruitful results have been achieved since Tao and Winkler's work in 2011. However, there is no any progress for the general case $\eta >0$
in the past ten years. In this paper, we analysed some commonly used research methods when $\eta =0$
, and found that these methods are completely unsuitable for situations where $\eta >0$
. By introducing some new forms of functionals, we reconstruct the relationship between the haptotactic term and the nonlinear diffusion term, and ultimately prove the global existence of weak solutions. This result improves and perfects a series of works previously presented in the literature.
Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$
, with $m>1$
, $q\in (0,\,1)$
and $\sigma =\sigma _c:=2(1-q)/ (m-1)$
is proved. Looking for radially symmetric solutions of the form\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]we show that there exists a unique exponent $\beta ^*\in (0,\,\infty )$
for which there exists a one-parameter family $(u_A)_{A>0}$
of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$
satisfying $f_A(0)=A$
and $f_A'(0)=0$
. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma \in (0,\,\sigma _c)$
.
In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$
): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$
$=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$
in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$
and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$
, where $A$
, $B$
, $L$
and $M$
are closed linear operators on a Banach space $X$
such that $D(A)\cap D(B)\subset D(M)\cap D(L)$
and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$
, $G$
and $F$
are bounded linear operators from $L^p([-2\pi,\,0];X)$
(respectively $B_{p,q}^s([-2\pi,\,0];X)$
) into $X$
, $u_t(\cdot ) = u(t+\cdot )$
and $u'_t(\cdot ) = u'(t+\cdot )$
are defined on $[-2\pi,\,0]$
for $t\in [0,\, 2\pi ]$
. We completely characterize the well-posedness of ($P_4$
) in the above two function spaces by using known operator-valued Fourier multiplier theorems.
In a ball $\Omega \subset \mathbb {R}^{n}$
with $n\ge 2$
, the chemotaxis system
\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]
is considered along with no-flux boundary conditions for $u$
and with prescribed constant positive Dirichlet boundary data for $v$
. It is shown that if $D\in C^{3}([0,\infty ))$
is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$
for all $\xi >0$
with some ${K_D}>0$
and $\alpha >0$
, then for all initial data from a considerably large set of radial functions on $\Omega$
, the corresponding initial-boundary value problem admits a solution blowing up in finite time.
We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtrationIf the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853].
$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$
We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:
with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:
uniformly on any compact set [δ, 1], δ ∈ (0, 1).
Considered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation
in a bounded domain Ω in ℝN (N≥2). We prove the regularity in the space L2p−2(Ω)∩ 
\begin{linenomath}
u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t)
\end{linenomath}
$D_0^2(\Omega,\sigma)$, and estimate the fractal dimension of the pullback attractor in L2(Ω).
