In this paper, we consider the following PDE-ODE system modelling cancer invasion with slow diffusion and ECM remodelling,

\[ \begin{cases} u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), \\ v_t=\Delta v+u-v, \\ \omega_t={-}v\omega+\eta \omega(1-u-\omega). \end{cases} \]

$\eta =0$

$\eta >0$

$\eta =0$

$\eta >0$

This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$. The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.

We consider the global existence for the following fully parabolic chemotaxis system with two populations

\[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]

$\Omega =\mathbb {R}^2$

$\Omega =B_R(0)\subset \mathbb {R}^2$

$\kappa _i,\chi _i>0,$

$i=1,2$

$0=\Delta v-v+u_1+u_2$

$0=\Delta v+u_1+u_2$

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3, …. The corresponding branching process is related to the semilinear partial differential equation for x ∈ ℝd, where A is the infinitesimal generator of a multiplicative semigroup and the pks, k = 2, 3, …, are nonnegative functions such that We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

A model of chemotaxis is analyzed that prevents blow-up of solutions. The modelconsists of a system of nonlinear partial differential equations for the spatial populationdensity of a species and the spatial concentration of a chemoattractant in n-dimensionalspace. We prove the existence of solutions, which exist globally, and are L∞-bounded onfinite time intervals. The hypotheses require nonlocal conditions on the species-inducedproduction of the chemoattractant.

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R1,this answers a question raised in [4].