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.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):