# Autonomous and non-autonomous semilinear parabolic problems

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

Partial differential equations are the natural mathematical models for many physical systems and processes: diffusion of heat, chemical reactions, the flow of fluids, and the dynamics of solids. As such they are used throughout the sciences, and in engineering.

However, these equations are often complicated, and it is unusual to be able to work out the solution of one of these equations "by hand" to end up with a convenient formula. More usually in actual applications these equations are approximated on a computer, and then an approximate "solution" is obtained this way.

Given how important these equations are in so many applications, it is natural for mathematicians to try to understand as much as they can about them, even if (or perhaps precisely because) we cannot in most cases write down their solution. Many of the questions mathematicians ask may even seem trivial at first sight, such as "does this equation have a solution?". But such questions can be hard to answer, or have an answer that leads to surprising new insights or points of view. As a simple illustration, one can ask whether the equation x^2=-1 has a solution. Here the answer depends on what you want the variable x to be; if x must be a real number then the answer is no (since any square is positive); but if you allow x to be a complex number the answer is yes, and by asking the question in the first place one can be lead to introduce a new concept (which in the case of complex numbers turned out to be incredibly powerful).

Another question at a similar level is, once we know that a solution does exist, whether or not this solution is unique. Again, this question can be more subtle that it first appears. Does the equation x^2=1 have a unique solution? No, since x=-1 and x=+1 are both solutions. But if we want a well-defined way to choose a particular solution, this equation does have a unique positive solution.

Similar questions, albeit in a more complicated setting, lie at the heart of the mathematical theory of partial differential equations. Given a model for which it is possible to show that there is indeed a unique solution, one can then go on to investigate further properties, e.g. whether the solution is positive, whether it decays to zero as it changes in times, whether it "blows up" in a finite time...

This project aims to look at a wide class of models that can be treated in a unified way; by studying the over-arching framework, rather than the particular models themselves, it becomes possible to prove general results and see how they arise from the underlying structure of the equations. Then, if we analyse various model equations within our framework, it becomes possible to work out which phenomena associated with these equations arise from their general class and which are more intrinsic to the particular equation we are studying.

The work will be done as a collaboration between Prof. Robinson, based at Warwick, and Prof. Rodriguez-Bernal, who will be visiting while on a sabbatical year from his home institution in Madrid. As well as our short-term goals here this visit will foster longer-term collaboration between us both and our research groups long after the year is over.

However, these equations are often complicated, and it is unusual to be able to work out the solution of one of these equations "by hand" to end up with a convenient formula. More usually in actual applications these equations are approximated on a computer, and then an approximate "solution" is obtained this way.

Given how important these equations are in so many applications, it is natural for mathematicians to try to understand as much as they can about them, even if (or perhaps precisely because) we cannot in most cases write down their solution. Many of the questions mathematicians ask may even seem trivial at first sight, such as "does this equation have a solution?". But such questions can be hard to answer, or have an answer that leads to surprising new insights or points of view. As a simple illustration, one can ask whether the equation x^2=-1 has a solution. Here the answer depends on what you want the variable x to be; if x must be a real number then the answer is no (since any square is positive); but if you allow x to be a complex number the answer is yes, and by asking the question in the first place one can be lead to introduce a new concept (which in the case of complex numbers turned out to be incredibly powerful).

Another question at a similar level is, once we know that a solution does exist, whether or not this solution is unique. Again, this question can be more subtle that it first appears. Does the equation x^2=1 have a unique solution? No, since x=-1 and x=+1 are both solutions. But if we want a well-defined way to choose a particular solution, this equation does have a unique positive solution.

Similar questions, albeit in a more complicated setting, lie at the heart of the mathematical theory of partial differential equations. Given a model for which it is possible to show that there is indeed a unique solution, one can then go on to investigate further properties, e.g. whether the solution is positive, whether it decays to zero as it changes in times, whether it "blows up" in a finite time...

This project aims to look at a wide class of models that can be treated in a unified way; by studying the over-arching framework, rather than the particular models themselves, it becomes possible to prove general results and see how they arise from the underlying structure of the equations. Then, if we analyse various model equations within our framework, it becomes possible to work out which phenomena associated with these equations arise from their general class and which are more intrinsic to the particular equation we are studying.

The work will be done as a collaboration between Prof. Robinson, based at Warwick, and Prof. Rodriguez-Bernal, who will be visiting while on a sabbatical year from his home institution in Madrid. As well as our short-term goals here this visit will foster longer-term collaboration between us both and our research groups long after the year is over.

### Planned Impact

The primary impact will be within academia, as detailed in the previous question.

We plan to engage with local secondary schools and investigate the possibility of talks to sixth formers, ideally as a "double act" that will demonstrate the international nature of mathematical research.

We plan to engage with local secondary schools and investigate the possibility of talks to sixth formers, ideally as a "double act" that will demonstrate the international nature of mathematical research.

## People |
## ORCID iD |

James Cooper Robinson (Principal Investigator) |

### Publications

Robinson J
(2018)

*Optimal existence classes and nonlinear-like dynamics in the linear heat equation in R d*in Advances in Mathematics
Robinson J
(2020)

*The heat flow in an optimal Fréchet space of unbounded initial data in R d*in Journal of Differential EquationsDescription | We have investigated unexpected behaviour in the heat equation. Although this is a classical, "well-understood" model, we have shown that if the initial condtions have significant amounts of heat "at infinity", i.e. at large distances from the original, then the equations can produce behaviour more typical of nonlinear models. Our research in this direction is continuing after the end of the funding period. |

Exploitation Route | Such unusual behaviour in a canonical model could inspire others to look for similar phenomena in other standard models. |

Sectors | Other |