Existence and uniqueness of a solution for the semilinear heat equation

Abstract

This thesis is about the existence and uniqueness of a solution for the semilinear heat equation of polynomial type. The extensive study of properties for these equations started off in the 1960s, when Hiroshi Fujita published his results that the existence and uniqueness of solutions depends critically on the exponent of the nonlinear term. In this thesis we expose some of the basic methods used in the theory of linear, constant coefficient partial differential equations. These considerations lay out the groundwork for the main result of the thesis, which is the existence and uniqueness of a solution to the generalized heat equation.

In Chapter 2 we expose the basics of functional analysis. We start off by defining Banach spaces and provide some examples of them. Then, we state the very useful Banach fixed point theorem, which guarantees the existence and uniqueness of a solution to certain types of integral equations. Next, we consider linear maps between normed spaces, with a focus on linear isomorphisms, which are linear maps preserving completeness. The isomorphisms prove to be very useful, when we consider weighted spaces. This is because for certain types of weights, we can identify the multiplication by weight with a linear isomorphism.

In Chapter 3 we introduce the Fourier transform, which is a highly useful tool for studying linear partial differential equations. We go through its basic mapping properties, such as, interaction with derivatives and convolution. Then, we consider useful spaces in Fourier analysis.

Chapter 4 is on the regular, inhomogeneous heat equation. A common method for deriving the solution to heat equation is formally applying the Fourier transform to it. This way we obtain a first order, linear ordinary differential equation, for which there is a known solution. The derived solution will serve as a motivator for how to approach the semilinear case. Also, in the end we will solve explicitly a slight generalization of the heat equation.

In Chapter 5 we prove the main result of this thesis: existence and uniqueness of a generalized solution for the semilinear heat equation. The methods we use in the proof are quite elementary in the sense that we do not need heavy mathematical machinery. We reformulate the generalized semilinear heat equation using an operator and show that it satisfies the conditions of the Banach fixed point theorem in a small, closed ball of a suitable Banach space.

We also include an appendix, in which we discuss differentiability properties of the generalized solution. It is possible to apply methods used in the proof of the generalized case to prove continuous differentiability. We provide some ideas on how one should approach the time differentiability of the solution by estimating the difference quotient of the integral operator.