In this article, we investigate singularities of solutions of anisotropic parabolic equations, and in particular the ones of the anisotropic porous medium equation. We focus on conditions for the removability of singularities for such solutions and compare analytically obtained removability results with observed solution behavior in numerical simulations.

For quasilinear elliptic equations, the problem can be formulated as follows. Let Ω be an open subset in ℝⁿ. The function u is defined in Ω\{x₀} and satisfied some quasilinear partial differential equation in Ω\{x₀}, i. e., except in the point x₀ where a singularity might lie. The removability problem consists of extending the function u to the entire domain Ω so that the extended function ũ satisfies the same quasilinear equation in Ω, and in finding conditions that guarantee the existence of the extension. If the extension of u to ũ is possible, we will say that the singularity in x₀ is removable.

Additionally, while dealing with equations of parabolic type like done in this article, singular initial data arise in a natural way. The problem statement remains the same, but it can be formulated in different ways: either as the question of a removable singularity or as the non-existence of a solution with a singularity.

The qualitative behavior of solutions to quasilinear elliptic and parabolic equations near the point singularity was investigated by many authors starting from the seminal paper of Serrin [1]. Further analysis of sufficient conditions for the removability of singularities of solutions has been made by many authors for different classes of nonlinear elliptic and parabolic equations, cf. [2] and the references therein. As for anisotropic elliptic and parabolic equations, their active research began recently. There are many scientists who presented fundamental results in the qualitative theory for such equations. Feo, Vázquez, Volzone, Song, Jian deal with questions about the existence of a fundamental solution [3], self-similar fundamental solutions [4, 5], existence and uniqueness of a bounded and continuous solution for equations with singular advections and absorptions [6, 7]. Skrypnik and his co-authors obtained removability results for the anisotropic versions of the porous medium equation and for the fast diffusion equation [8], the p-Laplacian equation [9] and doubly nonlinear anisotropic parabolic equations [10], including equations with an absorption term [11–16], etc.

The paper is organized as follows. In Section 2, we introduce the statement of the singularity problem for anisotropic parabolic equations. In Section 3, we provide the history of the removability problem for isotropic and anisotropic equations. In Section 4, we present the analytical results on the growth behavior of solutions near the singularities, which are validated and visualized by hands of numerical simulations in Section 5. The paper finishes with a resume and an outlook.

We study non-negative solutions to the anisotropic parabolic equation

where Ω_T = Ω × (0, T), Ω is a bounded open set in ℝⁿ with n≥2, which without loss of generality, contains the origin, i. e., x₀ = 0∈Ω, and where T with 0 < T < +∞ is a finite time. The initial condition is

and allows a concentrated essential weight in the origin.

Eq. (1) can be seen as a diffusion equation for the concentration u = u(t, x), and the diffusion parameters depend on the concentration u as well as on the direction in ℝⁿ via the different exponents m_i−1. The exponents m_i, which are not necessarily integers, have a strong physical background. In fact, they come from fluid dynamics in anisotropic media. If the conductivities of the media are different in different directions, the exponents m_i are different from each others [17].

In the special case m₁ = m₂ =...m_n = 1, Eq. (1) reduces to the isotropic heat equation. But for m_i>1, i = 1, …, n, the diffusion parameters tend to zero with decreasing concentrations. Thus, the diffusion process degenerates near zero concentrations. In this case, Eq. (1) is degenerate parabolic, and it is called an anisotropic porous medium equation [18]. On the other hand, for m_i < 1, i = 1, …, n, the equation is singular parabolic and called anisotropic fast diffusion equation [19].

As we see, the anisotropy of Eq. (1) is realized via the exponents m_i−1 in the concentration-dependent diffusion parameters umi-1. The case m_i>1 means that the diffusion strength increases with a growing positive concentration u, where m_i < 1 leads to a diffusion strength that increases up to infinity for decreasing u tending to 0. Therefore for small u and m_i < 1, we expect the faster leveling behavior the smaller u is in x_i-direction.

Here, we consider the case when anisotropy exponents are restricted by two conditions, namely first, a lower bound

and next, an upper bound depending on the mean of the exponents

As a first idea, conditions (3) and (4) mean that the exponents m_i might be commonly large but might not differ too much or be too small, comp. Section 4.2, where the admissible anisotropies are investigated in more detail. These conditions cover also the case where one part of the exponents m_i is greater than 1 and the other part m_i is less than 1.

Remark 1. In all known related publications, the cases of degenerate (m_i>1, i = 1, …, n) and singular (m_i < 1, i = 1, …, n) parabolic equations were considered independently from each other even in the isotropic case, i. e. for m₁ = m₂ =... = m_n = m. The used methods for proving the results depend on either the degenerate or singular character of equations.

Remark 2. Without loss of generality, we will assume that the point x0=0∈ℝn carries a singularity, otherwise we can make a change of variable by a simple translational shift.

Remark 3. Initial condition (2) can be written in the following way

In this case, it will be about the non-existence of solutions to the Cauchy problem with a singular initial condition, and not about the removability conditions.

Here, we are interested in solving the problem (1, 2) numerically and testing the analytical results from [8] which guarantee that the singularity at (0, 0) is removable.

The first theorem on removable singularities was obtained by Riemann. In his doctoral dissertation [1851, see Riemann [20]], he established the removability of an isolated singular point for a harmonic function of two real variables. In the general case, the necessary and sufficient condition of the removable singularity at the point x₀ for a harmonic function u in ℝn\{x0} has the form

Here

is the fundamental solution of Laplace's equation that exhibits the solution with the “minimal” singularity at x = x₀. It's easy to see how the condition (Equation 5) works if we expand the harmonic function into a series of spherical harmonics under the following form

where r, σ are the spherical coordinates in ℝn\{x0}, and ψ_i(σ), ψ~i(σ) spherical harmonics of degree n. If we assert that the condition (Equation 5) is satisfied, i.e., ũ(r, σ) = o(ε(r)) as r → 0, then the first term on the right side in Equation (7) is missing. It means that u is a harmonic function in the whole ℝⁿ. So this condition shows that there is no solution of Laplace's equation which is singular at the point x₀ and satisfies condition (Equation 5). It is obvious that the question of the removability of the singularity is conditioned by the growth of u near this point. If for example ũ(r,σ)=O(εb(r)) as r → 0, for some nonnegative integer b, then u admits an asymptotic expansion of the following form

and stays harmonic in ℝn\{x0}. Therefore, a crucial step in studying the singularity problem is the knowledge of an a priori estimate of u near the singularity.

Then for a long time, the only study of singularity problems dealt with linear equations and with radial solutions of Laplace's equation with nonlinear sources or absorptions. In fact, the first breakthrough is due to Serrin [1] who obtained the first general results on quasilinear equations. His precise condition on removability of singularity for nonnegative solutions of the p-Laplacian equation

reduces to

where ε(x−x₀) is the fundamental solution of the p-Laplacian equation and is described by the formula

Around 1980, the sharp development of the theory of nonlinear partial differential equations allowed another breakthrough in the study of nonradial singular solutions of Laplace's equations with nonlinear sources and absorptions. This was initiated by Gidas and Spruck [21], Lions [22] and Veron [23]. After this first period, many articles have been published taking into account the different aspects of the singularity problem for the above-mentioned equations and also for parabolic equations. We refer to the monograph by Veron [2] for an account of these results.

During the last decade, there have been growing interest and substantial developments in the qualitative theory of second-order anisotropic elliptic and parabolic equations e.g., [5, 24–30], in particular results for anisotropic porous medium equation can be found in Ciani and Henriques [31], Feo et al. [4], Henriques [32], Song and Jian [3], Song [6], and Song [7]. The study of these equations is complicated by the fact that a general qualitative theory for them has not been constructed, in addition, the explicit form of the fundamental solution is unknown in most of the cases. Therefore, the problem arises of obtaining precise conditions for the removability of the singularities for anisotropic elliptic and parabolic equations. Due to the fact that it is not possible to construct the fundamental solution of Equation (1) in an explicit form similar to Equations 6, 8, until recently it was not clear how to formulate the precise or at least sufficient condition for the removability of the singularity for the solution of this equation. This question was successfully solved in Namlyeyeva et al. [10], where is proved that the singularity at the point (x₀, t₀) with x₀ = 0∈ℝ and t₀ = 0 for the solution of the equation

is removable if the following condition holds

where is z(x,t)=(∑i=1n|xi−xi0|αi+tβ)−n, and the exponents are given by

with

The anisotropic doubly nonlinear parabolic (Equation 9) reduces to the anisotropic p-Laplacian evolution equation if m₁ = m₂ =... = m_n = 1. Further for p₁ = p₂ =... = p_n = 2, we obtain the degenerate case of Eq. (1). Other results on the removability of singularities for anisotropic equations concern special cases of Eq. (9) with absorption [11] and gradient absorption terms [12] and for anisotropic elliptic equations [9, 13–15]. But at this stage of the study, we are not interested in equations with additional terms.

Here, we present the numerical solution of Eq. (1) with the initial condition (Equation 2). It is solved by finite differences, and the singularity in the initial condition was replaced by a particular value conserving the integral. Of course, finite differences are not the ideal method to handle highly oscillating or highly changing values, and rather finite elements with their integrative aspect over each element would be appropriate.

But on the other hand, finite differences are a method which is not related to the removability condition in Eq. (10), which is an integral identity directly connected to the weak formulation of Eq. (1) and thus to finite elements. Therefore, we regard finite differences as a properly unbiased method. By the way, no qualitative difficulties occurred with the numerical solution in Matlab (as used here), Python, or Octave.

Figures 3, 4 show the time evolution of the concentrated initial value in Eq. (2) for n = 2. After a small time, the expected leveling behavior together with an anisotropy close to the origin is observable.

Next, we test the limit behavior given in Equation 11 as a removability condition. We compare the numerical solution u = u(t, x) for certain times t>0 with the estimate function v used in Eq. 12 to give an upper bound in the limit x → 0 and t → 0. Figure 5 shows the claimed small-o behavior of u, s. conditions (Equations 11, 12). Please remark that the comparison for t = 0 is not reasonable due to the vanishing initial values outside the origin. Although the small-o behavior of the estimate is numerically reproduced, the computed solution goes a little faster to 0 than the estimate. This coincides with the reformulation of the removability condition (Equation 12).

We observe that the quotient u/v of condition (Eq. 12) is indeed bounded and tends numerically to zero when (x, t) approaches the point (0, 0) carrying the singularity at the initial time. Furthermore, we see that the qualitative tendency observed in the numerical data u is well estimated by the analytical estimate v because the quotient approaches linearly zero in all directions.

Remark that no numerical artifacts are remarkable although the finite differences are a very rough numerical method. Together with the argument that the finite difference method is not biased as e. g. finite elements would be due to the condition in Eq. (10), which would make a non-removable singularity numerically not accessible at the same time, we rate the numerical simulation as a good validation and strong support of the power of the analytical estimate v in condition (Eq. 11).

We have shown that the removability conditions from [8] for the anisotropic porous medium equation and fast diffusion equation can be numerically reproduced and validated for the admissible anisotropies, whereat the conditions on feasible anisotropies allow not too large differences in the exponents m_i on the one hand but sufficiently multifaceted situation for modeling various physical situations.

Further research will focus on expanding the considerations of removability conditions to more general partial differential equations, e.g. anisotropic version of the evolution p−Laplacian equation. Another interesting question is whether some anisotropies with large differences between the exponents might lead to a comparable behavior of the solutions or whether some extenuated assertations about the growth and decay behavior of the solution can be found.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

MS: Conceptualization, Formal analysis, Writing – original draft, Writing – review & editing. DL: Conceptualization, Validation, Visualization, Writing – original draft, Writing – review & editing.