Stieltjes Bochner spaces and applications to the study of parabolic equations

Highlights

• A novel type of parabolic equation with Stieltjes time derivative is introduced.

• Suitable to study equations with impulses or lapses where they don't not evolve.

• Specific existence and uniqueness results rigorously demonstrated.

• Some realistic numerical examples related to population dynamics are provided.

Abstract

This work is devoted to the mathematical analysis of Stieltjes Bochner spaces and their applications to the resolution of a parabolic equation with Stieltjes time derivative. This novel formulation allows us to study parabolic equations that present impulses at certain times or lapses where the system does not evolve at all and presents an elliptic behavior. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples related to population dynamics.

Introduction

The main goal of this work is to analyze the existence of solution of the partial differential equation

$$\{\begin{array}{cc}{u}_g^{\prime }-\mathrm{\nabla }\cdot (k_1\mathrm{\nabla }u)+k_{2}u=f,\hfill & \text{in}[(0,T)\setminus C_g]\times \mathrm{\Omega },\hfill \\ u=0,\hfill & \text{on}(0,T)\times \partial \mathrm{\Omega },\hfill \\ u(0,x)=u_0(x),\hfill & \text{in}\mathrm{\Omega },\hfill \end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a domain with a smooth enough boundary ∂Ω and $u_g^{\prime }$ is the Stieltjes derivative in some Banach space V with respect to a left-continuous nondecreasing function $g:\mathbb{R}\to \mathbb{R}$. This is, given a function $u:[0,T]\to V$, we define for each $t\in [0,T]\setminus C_g$, $u_g^{\prime }(t)$ as the following limit in V in the case it exists:

$$u_g^{\prime }(t):=\{\begin{array}{cc}\underset{s\to t}{\lim }\frac{u(s)-u(t)}{g(s)-g(t)},\hfill & \text{if}t\notin D_g,\hfill \\ \frac{u(t^{+})-u(t)}{g(t^{+})-g(t)},\hfill & \text{if}t\in D_g,\hfill \end{array}$$

where

$$D_g=\{s\in \mathbb{R}:g(s^{+})-g(s)>0\}$$

and

$$C_g=\{s\in \mathbb{R}:g\text{ is constant on }(s-\epsilon ,s+\epsilon )\text{ for some }\epsilon \in {\mathbb{R}}^{+}\}.$$

The study of this type of derivatives and its application to the field of ODEs appears in [4], [5], [10], [11]. We use the notation established in previous works. We further assume that $k_1>0$ is a positive constant, $k_2\ge 0$, $u_0\in L^2(\mathrm{\Omega })$ and $f\in L_g^2([0,T],L^2(\mathrm{\Omega }))$, with $([0,T],{\mathcal{M}}_g,{\mu }_g)$ a suitable measure space associated to g [11]. It is important to mention that if $u:A\subset \mathbb{R}\to H$ is g-continuous for every $t_0\in A$ in the sense of:

$$\forall \epsilon >0\exists \delta >0:[t\in A,|g(t)-g(t_0)|<\delta ]\Rightarrow {\Vert u(t)-u(t_0)\Vert }_H<\epsilon ,$$

then f is constant in the same intervals as g [4, Proposition 3.2]. Moreover, continuity in the previous sense does not imply continuity in the classical sense, but if g is continuous at $t_0\in [0,T]$, then so is f [11]. Taking into account that g es left-continuous, we observe that the spaces of bounded g-continuous functions $\mathcal{B}{\mathcal{C}}_g([0,T],L^2(\mathrm{\Omega }))$ and $\mathcal{B}{\mathcal{C}}_g([0,T),L^2(\mathrm{\Omega }))$ are basically the same since any function in $\mathcal{B}{\mathcal{C}}_g([0,T],L^2(\mathrm{\Omega }))$ must be continuous at T.

Observe that the Stieltjes derivative is not defined at the points of $C_g$. The connected components of $C_g$ correspond to lapses when our system does not evolve at all and presents an elliptic behavior. The set $D_g$ of discontinuities of g correspond with times when sudden changes occur and which are usually introduced in the form of impulses. Finally, in the remaining set of times $[0,T]\setminus (C_g\cup D_g)$ the system presents a parabolic behavior and the different slopes of the derivator g (see [10]) correspond to different influences of the corresponding times, namely, the bigger the slope of g the more important the corresponding times are for the process. In a certain sense, system (1) can be considered as a degenerate parabolic system.

The main difficulty in the mathematical analysis of system (1) lies in the fact that we cannot consider the distributional derivative in time for defining the concept of solution. Thence, we will define the solution in terms of its integral representation and prove new Lebesgue-type differentiation results in order to recover the Stieltjes derivative g-almost everywhere in $[0,T]$. Results proven in the appendix of [2] suggest that it might be possible to define the concept of g-distributional derivative, thus proving the relationship between the g-absolute continuous functions and the $W_g^{1,1}$-type spaces. It is important to mention that in the case where $g(t)=t$, we recover the standard derivative, so all of the results that we will prove extend the classical theory.

In this work we will establish the basis of the mathematical analysis for system (1) as well as a first numerical approximation of its solution. In order to organize the contents of the paper, we will divide the work in the following sections: In Section 2 we will introduce the Stieltjes-Bochner spaces in which we will define the concept of solution. We will also prove new Lebesgue-differentiation-type results for the Stieltjes derivatives and some continuous injections. In Section 3 we will define the concept of solution of problem (1). In Section 4 we will prove an existence result for system (1) that generalizes some aspects of the classical theory of parabolic partial differential equations. Finally, in Section 5, we will present a realistic example and we will propose a numerical scheme. In this example we will have a parabolic-elliptical behavior, showing the advantage of considering derivatives of the Stieltjes type.