## Project Details

### Description

Finite-time blow-up (“explosion”) and finite-time quenching are well-known singular be- haviours of solutions to semilinear parabolic differential equations and (related) nonlinear Volterra integral equations. In the former case the value of the solution tends to infinity as the time approaches some finite value Tb (the blow-up time), while in the latter case the solution tends to some finite value its time derivative becomes infinite as the time approaches a finite value Tq (the quenching time). Blow-up or quenching may occur at a single point or at multiple points.

While the mathematical theory of blow-up and quenching is now well understood for parabolic differential equations and for Volterra integral equations (VIEs), this is not yet true for or- dinary or (parabolic) partial Volterra integro-differential equations (VIDEs) where the non- linear term is nonlocal (that is, it is given by a Volterra-type memory term reflecting the history of the solution up to the present time). Moreover, there are very few studies of computational schemes for approximating, e.g., the quenching time of nonlinear VIEs and semilinear ordinary or partial VIDEs.

The aims of this research project are

(i) to develop the mathematical theory of quenching for ordinary and parabolic VIDEs;

(ii) to design and analyse efficient and accurate numerical methods for approximating quenching solutions (and the quenching times) for VIEs and VIDEs;

(iii) to use the insight gained in (ii) to analyse and implement time-stepping methods for semilinear parabolic VIDEs whose solutions quench in finite time; and

(iv) to introduce numerical strategies for detecting quenching in solutions to VIEs and VIDEs.

While the mathematical theory of blow-up and quenching is now well understood for parabolic differential equations and for Volterra integral equations (VIEs), this is not yet true for or- dinary or (parabolic) partial Volterra integro-differential equations (VIDEs) where the non- linear term is nonlocal (that is, it is given by a Volterra-type memory term reflecting the history of the solution up to the present time). Moreover, there are very few studies of computational schemes for approximating, e.g., the quenching time of nonlinear VIEs and semilinear ordinary or partial VIDEs.

The aims of this research project are

(i) to develop the mathematical theory of quenching for ordinary and parabolic VIDEs;

(ii) to design and analyse efficient and accurate numerical methods for approximating quenching solutions (and the quenching times) for VIEs and VIDEs;

(iii) to use the insight gained in (ii) to analyse and implement time-stepping methods for semilinear parabolic VIDEs whose solutions quench in finite time; and

(iv) to introduce numerical strategies for detecting quenching in solutions to VIEs and VIDEs.

Status | Finished |
---|---|

Effective start/end date | 1/01/14 → 30/06/17 |

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