A remark on removable singularity for nonlinear convection–diffusion equation

Abstract

In this paper we study the following Cauchy problem:

$$u_t=u_{xx}+{\left(u^n\right)}_x\text{,}(x,t)\in \mathbb{R}\times (0,\infty ),u(x,0)=\delta \left(x\right)\text{,}x\in \mathbb{R}\text{,}$$

where $\delta \left(x\right)$ is a Dirac measure and $n\ge 0$. Its solution is called source-type solution. Such singular solution plays an important role in the development of theory of nonlinear parabolic equations. However, there seems not to be perfect answer to the research. Here we focus on whether there is a critical exponent $n_0$ such that when $n<n_0$ there exists unique source-type solution, while $n\ge n_0$ there is no source-type solution, and on what the singular expansions of source-type solutions at origin are. From a physical point of view, there are phenomena of the interactive effect between the diffusion and convection in a heat process, which is re-confirmed and described through mathematical analysis and numerical simulation. In addition, thanks to the entropy inequality, we get new proof of uniqueness and are able to extend our approaches to nonlinear parabolic-hyperbolic equations with Radon measure as initial datum.

Introduction

In this paper we study the following equation

$$u_t=u_{xx}+{\left(u^n\right)}_x\text{,}(x,t)\in S=\mathbb{R}\times (0,\infty )$$

with initial datum

$$u(x,0)=\delta \left(x\right)\text{,}x\in \mathbb{R}\text{,}$$

where $\delta \left(x\right)$ denotes the Dirac measure in $\mathbb{R},n\ge 0$ is a constant.

Eq. (1.1) arises in many disciplines of science. For instance, it describes a heat flow in certain material with temperature dependent on conductivity, dopant diffusion in semi-conductors, or the movement of a thin liquid under gravity, to name a few. It is well known that such a nonnegative solution of the problem (1.1)(1.2) is called source-type solution of Eq. (1.1) and it is a singular solution. Source-type solutions play an important role in the development of theory of nonlinear parabolic equations. During the past decades, Kamin  [1], Brezis and Friedman  [2], Zhao [3] and Brezis  [4] have studied the solvability of source-type solutions to the diffusion equation with absorptions, and Liu and Pierre  [5] have considered source-type solutions of the conservation laws respectively. However, equations with nonlinear convection is very different from equations with absorption or in conservation law. In this case, equations are of a mixed parabolic-hyperbolic type. they possess a nonlinearity and hyperbolic feature due to the nonlinear convection. They have many interesting properties as described in  [6], [7]. The solvability of source-type solution of this type of equations depends not only on diffusion but also on convection. Meanwhile, the methods presented in early papers cannot be generalized to deal with those situations.

Due to the complex mathematical nature of the aforementioned singular solutions, there seem to be no previous works on the nonexistence and the short time asymptotic behavior of the solutions. Although Escobedo, Vazquez and Zuazua  [6] have studied the existence of source-type solutions for the heat equation with convection, in our humble opinion they relied incorrectly on an unproven result to obtain their existence results, i.e., there is unique source-type solution of Eq. (1.1) when $n\ge 1$. In addition, in our previous work  [8] we did not provide a complete description for the singular behavior of source-type solutions for very short time. Hence there is need for further investigation on these important problems. In this paper, we develop a number of analytical techniques such as Moser’s iteration, an ODE method and scaling technique and use these techniques to obtain several accurate estimates for the approximating solutions and also create some special functions as subsolution and supersolution of Eq. (1.1) for comparison. The development of these techniques are based on our earlier works  [8], [9], [10], [11] and the methods developed in Chen and Zhang et al.  [12], [13]. As a result, we establish a nonexistence result and are able to describe the behavior of singularity expansion at the origin for very short time when such a singular solution exists (see Theorem A and Remark 1.1). Specifically we show the following:

• If $n$ is small, i.e.  $0<n<2$, the effect of convection is negligible as compared to diffusion and there exists unique source-type solution to Eq. (1.1), and for very short time its singularity at origin expands with the same behavior as the fundamental solution of the heat equation;

• If $n=2$, Eq. (1.1) is the so-called Burger’s equation and in this case an explicit self-similar source-type solution exists;

• If $n$ is larger, i.e.  $2<n<3$, the convection is stronger than diffusion. In this case, there exists unique source-type solution to Eq. (1.1) and for very short time its singularity at origin expands with the same behavior as the nonnegative fundamental entropy solution in the conservation law;

• If $n$ is large enough, i.e.  $n\ge 3$, the effect of convection dominate over diffusion. This leads to the fact that the diffusion and convection are not in coordination in the process, which makes impossible for the mass to concentrate at the origin in a very short time. Hence the singularity is removable and no source-type solutions of Eq. (1.1) exist.

As mentioned earlier, this paper emphasizes the interactive effects between the diffusion and the convection in the physical process and improves the results in Escobedo et al. [6] and in Lu et al.  [8]. Further, Escobedo et al. [6] does not discuss the existence of source-type solution for $0<n<1$ and the nonexistence of such singular solution for $n\ge 3$. In view of no optimal estimates, they were unable to show the existence result in the cases $0<n<1$ and the nonexistence results in the case $n\ge 3$. In particular, they introduced the Cauchy problem which is equivalent to equation $v_t=v_{xx}+{\left(v_x\right)}^n$ with Heaviside’s function as initial datum, and claimed that there exists a global solution to the problem without a strict mathematical derivation. Based on the derivation they concluded the existence of source-type solution of (1.1). The latter, however, is not perfect in describing the behavior of the singularity expansion at the origin. In this paper we improve the results in  [6], [8] by providing a rigorous analysis on the existence and nonexistence of source-type solutions and a complete description of their short time singular behavior. Furthermore, we conduct some numerical simulation to confirm the nonexistence results, i.e., when $n\ge 3$ the velocity $n{u}^{n-1}{u}_x$ is high enough at the origin, for a very short time the mass cannot be concentrated at the origin. It is worth mentioning that the existence for such a singular solution only requires to check the singularity of approximating solutions in the direction of $t$ but the nonexistence requires to check the size of the approximating solutions in both the directions $x$ and $t$. On the other hand, we recently showed that the above problem has no global solutions when $n\ge 3$ (see  [14]).

Traditionally, to establish the uniqueness of source-type solution we must estimate the solutions of the adjoint problems. Since Eq. (1.1) is of a mixed parabolic-hyperbolic type and the source-type solution is singular, it is extremely difficult to obtain uniformly prior estimates. Thanks to the entropy condition in conservation law (see [5]), we are able to generalize Kruzkov’s inequality  [15] for the hyperbolic equation to that for the parabolic-hyperbolic equation and to provide new proofs for the uniqueness of source-type solution to Eq. (1.1). The method we developed to prove the uniqueness in this paper can be applied to the Cauchy problem of nonlinear parabolic-hyperbolic equations with a Radon measure as initial datum as well as with Dirac’s measure, which extends the results in this area.

Definition 1

A function $u(x,t)$ defined in $S_T=\mathbb{R}\times (0,T)(T>0)$ is called source-type solution of Eq. (1.1), if and only if:

• $u(x,t)$ is nonnegative, continuous in $\overline{S_T}\setminus \left\{(0,0)\right\}$ and bounded in $\overline{S_T^{\tau }}=\mathbb{R}\times [\tau ,T]$ for any $\tau$: $0<\tau <T$.

• Eq. (1.1) holds in the distribution sense, i.e. 

  $${\int }_0^{\infty }{\int }_{-\infty }^{\infty }(u{\phi }_t+u{\phi }_{xx}-u^n{\phi }_x)dxdt=0\text{,}$$

  for any $\phi (x,t)\in C_0^{2,1}\left(S_T\right)$ with $\phi (x,T)=0$ for $x\in \mathbb{R}$.

• As $\tau \to 0^{+}$,

  $$u(x,t)\to \delta \left(x\right)$$

  in the sense of distribution, i.e. 

  $$\underset{\tau \to 0^{+}}{lim}{\int }_{-\infty }^{\infty }u(x,\tau )\eta \left(x\right)dx=\eta \left(0\right)$$

  for any $\eta \left(x\right)\in C_0^{\infty }\left(\mathbb{R}\right)$.

Definition 2

A function $u(x,t)$ defined in $S_T$ is called a supersolution (or subsolution) of Eq. (1.1), if and only if:

• $u(x,t)$ is nonnegative, continuous in ${\bar{S}}_T\setminus \left\{(0,0)\right\}$ and bounded in $S_T^{\tau }$ for any $\tau :0<\tau <T$.

• The integral inequality

  $${\int }_0^{\infty }{\int }_{-\infty }^{\infty }(u{\phi }_t+u{\phi }_{xx}-u^n{\phi }_x)dxdt\le 0(\text{or}\ge 0)$$

  holds for any nonnegative $\phi \in C_0^{2,1}\left(S_T\right)$ with $\phi (x,T)=0$ for $x\in \mathbb{R}$.

The main result of this paper is the following theorem:

Theorem A

There exists a critical number $n_0=3$, such that if exponent $n<n_0$ in convection term, there exists unique source-type solution of Eq.   (1.1), while such a solution does not exist if $n\ge n_0$.

Theorem B

When $n\ge n_0$, there exists no self-similar source-type solution of Eq.   (1.1).

Remark 1.1

Let $u(x,t)$ be source-type solution of Eq. (1.1). If $0<n<2$, then

$$t^{\frac{1}{2}}|u(x,t)-w(x,t)|\to 0$$

as $t\to 0^{+}$ uniformly with respect to $x$ in any domain

$$\{(x,t)\in S\mid \left|x\right|\le A{t}^{\frac{1}{2}}\}$$

for some constant $A>0$, where $w(s,t)=\frac{1}{2\sqrt{\pi t}}{e}^{-\frac{x^2}{4t}}$; If $n=2$, then the problem (1.1)(1.2) can be solved by

$$u(x,t)=t^{-\frac{1}{2}}{e}^{-\frac{x^2}{4t}}/(\frac{2\sqrt{\pi }}{e-1}+{\int }_{-\infty }^{\frac{x}{\sqrt{t}}}{e}^{-\frac{1}{4}{s}^2}ds)\text{.}$$

If $2<n<3$, then

$$\underset{\left|x\right|\le A{t}^{\frac{1}{n}}}{\mathrm{esssup}}{t}^{\frac{1}{n}}|u(x,t)-\tilde{w}(x,t)|\to 0$$

as $t\to 0^{+}$ for any $A>0$, where

$$\tilde{w}(x,t)=\{\begin{array}{cc}{\left(\frac{-x}{nt}\right)}^{\frac{1}{n-1}}\text{,}\hfill & \frac{-n}{{(n-1)}^{\frac{n-1}{n}}}{t}^{\frac{1}{n}}\le x<0\text{,}\hfill \\ 0\text{,}\hfill & x<\frac{-n}{{(n-1)}^{\frac{n-1}{n}}}{t}^{\frac{1}{n}}\text{or}x\ge 0\text{.}\hfill \end{array}$$

Furthermore as $t\to 0^{+},\bar{u}(s,t)\to \bar{W}\left(s\right)$ in the sense of graph in ${\mathbb{R}}^2$, where $x=s{t}^{\frac{1}{n}},\bar{u}(s,t)=t^{\frac{1}{n}}u(x,t),\bar{W}\left(s\right)=t^{\frac{1}{n}}\tilde{w}(x,t)$.

Remark 1.2

The situation of several dimensional case is different from one-dimensional case. As far as we know, the former case has no perfect answer yet. Although Escobedo et al.  [16] studied the question $u_t=△u+\frac{\partial u^n}{\partial x_N}$ for $N>1$ and claimed the critical number $n_0=\frac{N+1}{N-1}$. When $1<n<n_0$, there is unique source-type solution of the equation, while there is no source-type solution when $n\ge n_0$. Obviously the equation with the convection $\vec{\beta }(x,t)\cdot \nabla u^n$ and $\vec{\beta }$ constant vector, one can reduce it to Escobedo’s case by rescaling. Generally, the equation with the convection $\vec{\beta }(x,t)\cdot \nabla u^n$ cannot be reduced to such case. The authors  [10] studied the convection coefficient $\vec{\beta }(x,t)$ with all positive components and cannot be reduced. They claimed the critical number $n=\frac{N+2}{N}$. When $N=2$, Escobedo’s case has the same critical number as Eq. (1.1) in one-dimension. In fact, one can integrate Escobedo’s case for $x_1$ and deduce to one-dimensional case readily. The mathematical result coincides with physics. But we do not know what happen to the equation with the convector ${\sum }_{i<N}{b}_i(x,t)\frac{\partial u^n}{\partial x_i},b_i>0,1<i<N$. There seem to be no known results.

The organization of the paper is as follows. In Section  1, we introduce the background of the works concerned and state our main results of the paper. In Section  2, we introduce some basic estimates for source-type solution of Eq. (1.1)[8], [9], [11] which we will use as several tools in following sections of the paper. Section  3 is devoted to discussion of the self-similar source-type solution to Eq. (1.1) if $n\ge 3$ (Theorem B). In Sections  4 Uniqueness, 5 Nonexistence, we establish the uniqueness and nonexistence results (Theorem A) respectively. Final section, we demonstrate the conclusions of existence and nonexistence by numerical results. As for the existence result of Theorem A and Remark 1.1, the readers can refer to  [8], [17]. Here the details are omitted.