Qualitative theory for nonlinear non-local reaction-diffusion equations

Ladas, Nikolaos M. (2024). Qualitative theory for nonlinear non-local reaction-diffusion equations. University of Birmingham. Ph.D.

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Abstract

This Thesis is concerned with the qualitative properties of nonlinear non-local reaction diffusion equations. We begin by establishing minimum and comparison principles for solutions to inequalities involving the integro-differential operators

$P[u]=\sum_{i,j=1}^n a_{ij}\partial_{x_ix_j} u+ \sum_{i=1}^n {b}_i \partial_{x_i} u+cu+{d}Ju-\partial_t u,\quad\text{on } \Omega_T,$

and

$Q[u]=\sum_{i,j=1}^n a_{ij}\partial_{x_ix_j} u+ f(\cdot,\nabla u,u,Ju)-\partial_t u,\quad\text{on } \Omega_T$

respectively, with $Ju$ denoting the convolution of $u$ with an appropriately chosen integral kernel $\varphi$ . The minimum and comparison principles are established under a variety of assumptions on the coefficients $a_{ij}$ , $b_i$ , $c$ , $d$ and growth/decay rates of $u$ . Next, we demonstrate that the Cauchy problem associated with

$\partial_t u= \Delta u+ f(u,Ju),\quad\text{on } \Omega_T,$ ,

is well-posed, locally in time, when the nonlinear non-local term $f$ is locally Lipschitz continuous. Additionally, we prove the existence of solutions when $f$ is locally Hölder continuous (obtaining the existence of maximal and minimal solutions when $f$ is assumed to be non-decreasing with respect to $Ju$ ). Afterwards we treat the non-local analogue to a problem arising from fractional-order autocatalysis (by taking $f$ ( $u$ , $Ju$ ) $=$ ( $Ju$ ) $^{\rho} _+$ , for $p$ $\epsilon$ ( $0$ , $1$ )) and show its well-posedness (locally in time). We accompany our analysis with numerical simulations, demonstrating the conditional converges of the finite difference scheme, and large- $t$ asymptotics. Finally, we consider potential generalisations and extensions of the results presented in the text.