Fundamental solution for Cauchy initial value problem for parabolic PDEs with discontinuous unbounded first-order coefficient at the origin. Extension of the classical parametrix method

TL;DR

This paper establishes the existence and estimates of a fundamental solution for a class of parabolic PDEs with discontinuous, unbounded coefficients at the origin, extending Levi's classical parametrix method.

Contribution

It extends the classical parametrix method to handle parabolic PDEs with discontinuous, unbounded coefficients at the origin, proving fundamental solution existence and estimates.

Findings

Existence of fundamental solution for PDEs with discontinuous coefficients

Non-asymptotic, rapidly decreasing estimates at infinity

Extension of Levi's classical parametrix method

Abstract

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method provided by E.E. Levi.

Equations

$$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}+\frac{b(t,x)}{|x{|}^{\gamma }}\frac{\partial u}{\partial x},$$

$$\underset{t\to s+}{\lim }u(t,x)=f(x),u=u(t,x)=u[f](t,x),s\ge 0,$$

$$L[b{]}_{t,x}=L_{t,x}\stackrel{def}{=}\frac{\partial }{\partial t}-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}-\frac{b(t,x)}{|x{|}^{\gamma }}\frac{\partial }{\partial x}.$$

$$\underset{t\to s+}{\lim }{\int }_{\mathbb{R}}p(t,x,s,y)f(y)dy=f(x),x\in \mathbb{R},$$

$$p(t,x,s,y)\ge 0;{\int }_{\mathbb{R}}p(t,x,s,y)dy=1$$

$$p(t,x,s,y)={\int }_{\mathbb{R}}p(t,x,v,z)p(v,z,s,y)dz,0<s<v<t.$$

$$u(t,x)={\int }_{\mathbb{R}}p(t,x,s,y)f(y)dy.$$

$$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}+\frac{b(t,x)}{|x{|}^{\gamma }}\frac{\partial u}{\partial x}+g(t,x),t>s,$$

$$u(t,x)={\int }_s^{t}d\theta {\int }_{\mathbb{R}}p(t,x,\theta ,y)g(\theta ,y)dy.$$

$$Z(t,x,s,y)=Z_{t-s}(x-y):=(2\pi (t-s){)}^{-1/2}\exp \left\{-\frac{1}{2}\frac{(x-y{)}^2}{(t-s)}\right\},$$

$$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}.$$

$$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}+b_1(t,x)\frac{\partial u}{\partial x},$$

$$|b_1(t,x)|\le C\left(1+|x{|}^{\beta }\right),0<C,\beta =\mathrm{const}<\infty .$$

$$K=K[f]:=\underset{t,x}{\sup }|b(t,x)|=||b(\cdot ,\cdot )|{|}_{C((0,\infty )\otimes \mathbb{R})}<\infty .$$

$$p(t,x,s,y)=Z_{t-s}(x-y)+{\int }_s^t{\int }_R{Z}_{t-r}(x-z)\Phi (r,z,s,y)drdz,$$

$$\Phi (t,x,s,y)={\Phi }_1(t,x,s,y)+{\int }_s^t{\int }_R{\Phi }_1(t,x,r,z)\Phi (r,z,s,y)drdz$$

$${\Phi }_1(t,x,s,y)=\frac{b(t,x)}{|x{|}^{\gamma }}\frac{\partial }{\partial x}{Z}_{t-s}(x-y).$$

$$f\ast g(t,x,s,y)\stackrel{def}{=}{\int }_s^{t}dr{\int }_{R}f(t,x,r,z)g(r,z,s,y)dz,$$

$$\Phi (t,x,s,y)=\sum _{m=1}^{\infty }{\Phi }_m(t,x,s,y),$$

$${\Phi }_2(t,x,s,y)={\Phi }_1(t,x,s,y)\ast {\Phi }_1(t,x,s,y)$$

$${\Phi }_{m+1}(t,x,s,y)={\Phi }_1(t,x,s,y)\ast {\Phi }_m(t,x,s,y),m=2,\dots ,$$

$$p=Z+\sum _{m=1}^{\infty }Z\ast {\Phi }_m.$$

$${\Phi }^{(2)}(t,x,s,y)=\sum _{m=2}^{\infty }{\Phi }_m(t,x,s,y),$$

$$\left|\frac{\partial }{\partial x}{Z}_{t-s}(x-y)\right|\le (2\pi e\delta {)}^{-1/2}(t-s{)}^{-1}\times \exp \left\{-\frac{(1-\delta )(x-y{)}^2}{2(t-s)}\right\},$$

$${\Phi }_1(t,x,s,y)\le \frac{K[f]}{|x{|}^{\gamma }}(2\pi e\delta {)}^{-1/2}(t-s{)}^{-1}\times \exp \left\{-\frac{(1-\delta )(x-y{)}^2}{2(t-s)}\right\}.$$

$$\underset{\Delta >0}{\min }\left(A{\Delta }^{1-\gamma }+B{\Delta }^{-\gamma }\right)=\kappa (\gamma )A^{\gamma }{B}^{1-\gamma },$$

$$\kappa (\gamma )={\left(\frac{\gamma }{1-\gamma }\right)}^{1-\gamma }+{\left(\frac{\gamma }{1-\gamma }\right)}^{-\gamma },\gamma \in (0,1).$$

$${\int }_{\mathbb{R}}\exp \left\{-0.5\left(\frac{(x-z{)}^2}{{\sigma }_1^2}+\frac{(z-y{)}^2}{{\sigma }_2^2}\right)\right\}dz=(2\pi {)}^{1/2}{\sigma }_1{\sigma }_2{\sigma }_3^{-1}\exp \left\{-0.5\frac{(x-y{)}^2}{{\sigma }_3^2}\right\},$$

$${\sigma }_3^2={\sigma }_1^2+{\sigma }_2^2.$$

$$J=J(x,t,y,s):={\int }_{\mathbb{R}}|z{|}^{-\gamma }dz\exp \left\{-0.5\left(\frac{(x-z{)}^2}{t-r}+\frac{(z-y{)}^2}{r-s}\right)\right\}.$$

Full text

1 Parthenope University of Naples, via Generale Parisi 13,

Palazzo Pacanowsky, 80132, Napoli, Italy.

e-mail: mara.formica@uniparthenope.it

2 Department of Mathematics and Statistics, Bar-Ilan University,

59200, Ramat Gan, Israel.

e-mail: eugostrovsky@list.ru

e-mail: sirota3@bezeqint.net

Key words and phrases:

Partial Differential Equation of parabolic type, parametrix method, fundamental solution, degenerate diffusion, Gamma and Beta- functions, Markov random process, transfer density of probability for diffusion random processes, infinitesimal operator, generalized Mittag-Leffler function, Chapman-Kolmogorov equation, Neumann series, Hölder’s continuity, Volterra’s integral equation.

2010 Mathematics Subject Classification: 35A08, 35K15 35K20.

1. Definitions. Notations. Statement of problem.

We consider in this article the Partial Differential Equation (PDE) of (uniform) parabolic type

[TABLE]

where $u=u(t,x)$, $t\geq 0,\ x\in\mathbb{R},\ \ \gamma=\rm const\in(0,1)$ and the coefficient $b=b(t,x)$ is a continuous bounded function such that \ b(t,0)\neq 0,\ together with the initial value problem (Cauchy statement)

[TABLE]

where $f(\cdot)$ is a measurable function satisfying some growth condition at $|x|\to\infty$; for the definiteness one can suppose its continuity and boundedness, i.e. $\operatorname*{ess\,sup}_{x}|f(x)|<\infty$.

Briefly, \ L_{t,x}u=L[b]_{t,x}u=0,\ where \ L[b]_{t,x}\ is the linear parabolic partial differential operator of the form

[TABLE]

Let us recall the classical definition.

Definition 1.1**.**

The (measurable) function $\ p=p(t,x,s,y),\ 0<s<t;\ x,y\in\mathbb{R}^{2}$ is said to be a Fundamental Solution (F.S.) for the equation (1.1), subject to the initial condition (1.2), iff for all the fixed values \ (s,y),\ 0<s<t,\ x,y\in\mathbb{R}^{2}\ it satisfies the equation (1.1) and, for all the bounded continuous functions $f=f(x)$,

[TABLE]

if, of course, there exists.**

In this case the F.S. $p(t,x,s,y)$ may be interpreted as the transfer density of probability for diffusion random non-homogeneous in the time, as well as in the space, Markov process, see e.g. [2], [3]. As a consequence:

[TABLE]

and satisfies the well-known Chapman-Kolmogorov equation

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On the other words, the (unique) solution $u=u(t,x)$ of the Cauchy problem (1.1), (1.2), in the whole space $\mathbb{R}$, may be written as follows

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The solution of the non-homogeneous parabolic PDE of the form

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with zero initial condition $u(s,x)=0$, may be written, as ordinary, as follows

[TABLE]

For instance, the well-known Gauss function

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$0<s<t<\infty,\ x,y\in\mathbb{R}$, is the Fundamental Solution (F.S.) for the ordinary heat equation

[TABLE]

Our claim is to prove the existence of F.S. for the equation (1.1) with the initial condition (1.2), under suitable conditions.

The existence of F.S. for strictly parabolic PDE with smooth coefficients, for instance, bounded and belonging to the Hölder space, is explained in many works, e.g. [2], [6], [4], [12], [13], [17], [24], [25], [26], [27]. The same problem for parabolic systems is considered in [9], [10], [38]. The case of parabolic equations with discontinuous but bounded coefficients is investigated in [4], [16], [19], [21], [34], [39] etc.

We will essentially follow the interesting article of Thomas Deck and Susanne Kruse [4], where the authors considered the case of the equation

[TABLE]

in which the coefficient \ b_{1}=b_{1}(t,x)\ is Hölder continuous in $x$ and may be unbounded at $\ x\to\pm\infty$, in the following sense

[TABLE]

The uniqueness of the F.S. for parabolic PDEs is proved, e.g. in [5], [6], [13], [19], [24], [34].

In the probability theory, more precisely, in the theory of diffusion random processes, the F.S. $p(t,x,s,y)$ represents the density of transition of probability. The equations of the form (1.1) appears in particular in [30], [32]; see also [2], [3].

2. Main result: Existence of the Fundamental solution. Upper and lower estimates.

Theorem 2.1**.**

Suppose that the coefficient $b=b(t,x)$ in the equation (1.1) is measurable and bounded, i.e.

[TABLE]

Then there exists a Fundamental Solution $p(t,x,s,y)$ for the equation (1.1).

Proof.

A. We will follow the proof in [4], herewith we retain analougus notations, so that we state that the F.S. is given by

[TABLE]

where the function \ \Phi=\Phi(t,x,s,y)\ is a solution of the following Volterra equation

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with weak singular kernel

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The solution of the equation (2.3) may be obtained, as ordinary, by means of Neumann series. As in [4] we define the two-dimensional convolution

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then

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where

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and sequentially

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so that the F.S. has the form

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Introduce also the function

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It remains to ground the convergence of the Neumann series (2.6). We will need some auxiliary estimates.

B.  Here and henceforth let $\delta$ be an arbitrary constant,   $0<\delta<1$. The following estimate

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holds, and consequently

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C.  Let \ A,B=\rm const>0;\ then

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where

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D.  Recall the following identity which is well known in the probability theory, in the theory of Gaussian distribution:

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where $\sigma_{1},\ \sigma_{2},\ \sigma_{3}>0$ satisfies

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E.  Let us evaluate the following important auxiliary parametric integral which will be used after

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We split this integral $J$ into two ones $J=J_{1}+J_{2}$, where correspondingly

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Evidently,

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For the second integral we have

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Using equality (2.11) we get

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so that

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and by (2.10)

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where

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F.  Let us consider the following more general parametric integral

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where $\delta=\rm const\in(0,1)$. We have, after a change of variables,

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or equivalently

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and consequently

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G. The first term \ \Phi_{1}(\cdot)\ in (2.6) is estimated in (2.9). Let us estimate the second one \ \Phi_{2}.\ We have, after simple calculations,

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and, applying the estimate (2.13), we have

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where

[TABLE]

and $B(\cdot,\cdot)$ and \ \Gamma(\cdot)\ are respectively the ordinary Beta and Gamma functions. Note that the last integral is finite as long as \ \gamma\in[0,1).\

Evidently,

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H. Let us investigate a more general case. Define the next functions

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[TABLE]

where

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and define the convolution

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We deduce analogously

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so that

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More generally, if \ \delta=\rm const\in(0,1),\ then

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[TABLE]

[TABLE]

[TABLE]

I. By means of induction and using (2.18) we get

[TABLE]

where the constant \ \gamma\in(0,1)\ is given,  \ \delta\in(0,1)\ is arbitrary.

J. Recall briefly the classical definition of the so-called (generalized) Mittag-Leffler function ([37], [29])

[TABLE]

Here \ \alpha=\rm const>0,\ \beta\geq 0.\ We modify slightly this definition. Namely, we introduce the function

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We need to use only real non-negative values $z\in[0,\infty)$, but the expression in (2.20) is also defined for all the complex values \ z.\

The asymptotic as well as non-asymptotic behavior of these functions, as \ |z|\to\infty\, has been investigated in [37], see also [8].

We introduce also a slight modification of this function. Denote

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and alike

[TABLE]

Here \ c,\lambda,\mu=\rm const>0,\ but we need only the values \ \lambda\ in the set \ (0,1).\

Evidently,

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In order to obtain the asymptotic behavior of this function as \ z\to\infty,\ we will use the methods in the book of V.N. Sachkov ([35, chapters 2,3]), which is in turn the discrete analog of the classical saddle-point method.

Indeed, denoting \ \Lambda=|\ln(1-\lambda)|,\ \lambda\in(0,1)\, when z\geq e^{e}\Lambda\

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and alike

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Remark 2.1**.**

Notice that the function $\psi=\psi(z)$, introduced in (2.21), decreases rapidly at $z\to\infty$. Namely, let $B$ be an arbitrary positive constant, then

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Further, let us choose \ \lambda=\lambda(\gamma)=1-\gamma,\

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[TABLE]

and

[TABLE]

where evidently

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It follows immediately from this definition and from the relation (2.6) that

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where

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[TABLE]

K.  Ultimately, let now estimate the F.S. \ p=\ p(t,x,s,y)\ for the source equation (1.1). Define again the following variables

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[TABLE]

then, from the relations (2.2), (1.9), (2.26), we have

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The last expression may be simplified as follows. Suppose

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the opposite case is trivial. Define a new function

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and

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We deduce, under the above conditions, the following non-asymptotic upper estimate for the fundamental solution of the equation (1.1)

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Now we give the lower estimate. Suppose in addition that

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As long as

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we deduce, after some computations and under the same conditions, that for some positive value \ B_{1}\

[TABLE]

$\Box$

It is interesting to note that, for the parabolic PDE with smooth coefficients, the bilateral estimates for the Fundamental Solution has a Gaussian form (see, e.g. [1], [10]).

3. Solution of the equation (1.1) by means of the fundamental solution.

Theorem 3.1**.**

Suppose, in addition to the conditions of Theorem 2.1, that the continuous initial function $f=f(x)$ in the equation (1.1) has no more than polynomial growth at infinity:

[TABLE]

Then the (unique) solution $u=u(t,x)$ of the equation (1.1), with initial Cauchy condition (1.2), has the representation

[TABLE]

where $p$ is the F.S. of (1.1).

Proof. The relation (1.4) is obvious. The (uniform) convergence of the integral in (2.7) follows immediately from the condition (3.1) and estimation (2.27), as well as the initial condition (1.2).

Further, it is easy to verify, analogously to the proof of Theorem 2.1, that the convergence series in (2.7) hods true also for the partial derivatives $\ \partial p/\partial t,\ \partial p/\partial x,$ and for $\ \partial^{2}p/\partial x^{2}.$

See also [22], [28], [31].

This completes the proof. $\Box$

4. Concluding remarks.

A. The method here presented perhaps may be generalized on the equations of the form

[TABLE]

where \ M=M(x),\ x\in\mathbb{R}\ is some continuous slowly varying at the origin function.

B. The previous method may be extended to the more general $d$-dimensional case, d=2,3,\ldots\, of the parabolic differential equation of the form

[TABLE]

\ u=u(t,\vec{x}),\ with ordinary Cauchy initial condition

[TABLE]

where all the coefficients \ \{a_{i,j}(\cdot,\cdot),\ b_{i}(\cdot,\cdot),\ c_{i}(\cdot,\cdot))\}\ are Hölder continuous, the matrix \ \{a_{i,j}(\cdot,\cdot)\}\ is symmetric, positive definite and bilateral bounded:

[TABLE]

[TABLE]

[TABLE]

C. An open and interesting, by our opinion, problem: investigate the existence of F.S. and its properties for the initial problem for the parabolic differential equation with variable coefficients of the (very singular) form

[TABLE]

The operator \ L_{B}\ is the infinitesimal operator for the so-called Bessel’s non-homogeneous random process.

The case \ b(t,x)=\rm const\ was considered in [30], where was shown some probabilistic applications. A particular cases was represented in the monograph [33], see also [36].

A very interest application of the multivariate Bessel processes is described in the recent article [40], see also an article [7].

D. Perhaps, one can study also the quasi-linear parabolic PDE of the form, for instance,

[TABLE]

with initial condition (1.2), or more generally

[TABLE]

where

[TABLE]

[TABLE]

under appropriate conditions, e.g. with the symmetrical positive definite bounded and Hölder’s continuous matrix \ \{a_{i,j}(\cdot,\cdot,\cdot,\cdot)\}\ as well as the coefficients \ \{b_{i}(\cdot,\cdot,\cdot)\},\ c(\cdot,\cdot,\cdot)\ and with the same initial condition (1.2).

One can consider, for instance, following the authors of articles  [11],  [18],  [20],  [23], the iteration procedure (recursion) consisting only of the linear equations

[TABLE]

with the initial condition relative to the number of iterations $n$

[TABLE]

Evidently, if there exists the limit $u(t,x):=\lim_{n\to\infty}u_{n}(t,x)$ in the space \ C_{\rm loc}^{1,2}(\mathbb{R}_{+},\mathbb{R}^{d}),\ it satisfies the equation (4.4), (4.5) with initial condition (1.2).

Acknowledgement. The first author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project “sostegno alla Ricerca individuale”.