Finite extinction for a doubly nonlinear parabolic equation of fast diffusion type

Purpose – The purpose of this paper is to find a doubly nonlinear parabolic equation of fast diffusion in a bounded domain. Design/methodology/approach – For positive and bounded initial data, the authors study the initial zero-boundary value problem. Findings – The findings of this study showed the complete extinction of a continuous weak solution at a finite time. Originality/value – The extinction time is studied earlier but for the Laplacian case. The authors presented the finite extinction time for the case of p-Laplacian.

First of all, we will recall some fundamental results for the Eqn (1.1).
In the case p ¼ 2, the Eqn (1.1) becomes the so-called porous medium equation or the plasma equation with the Sobolev critical exponent q þ 1 ¼ 2n n − 2 . The global existence and continuity of a weak solution of (1.1) in the case p ¼ 2 is proved in [1][2][3]. In particular, the extinction at a finite time of a continuous weak solution is shown in [4,5]. The positivity of the unique weak solution is demonstrated in [6] and the decay estimation has presented in [6,7]. The asymptotic behavior at the extinction time is also studied (see [8,9]). The regularity results for the porous medium equations and the p-Laplace equations are also established and are the fundamental theory for the degenerate and singular parabolic equations (see [10,11]). Here we will consider a doubly nonlinear equation (1.1) with the p-Laplacian, the porous medium operator and the Sobolev critical exponent and study the positivity, boundedness and finite time extinction of a weak solution of Eqn (1.1). The H€ older regularity of a weak solution to Eqn (1.1) in homogeneous case q þ 1 ¼ p accomplished in [12]. On the other hand, in the nonhomogeneous case that q þ 1 is not equal to p, the regularity of a weak solution is remained to be settled.
A doubly nonlinear parabolic equation, like the one considered above, has been studied before, and the global existence of a weak solution is proved in [13,14] and, for a positive, bounded initial data in the Sobolev space W 1;p 0 ðΩÞ, the boundedness, the expansion of positivity and regularity of a weak solution to Eqn (1.1) are accomplished in [15] in the Sobolev critical case that q þ 1 ¼ np n − p . The finite time extinction is also shown in [13]. These results are the starting point of our study in the paper and thus, we will recall these results later. Now we will explain in details what we mean by complete extinction for solutions of problem in Eqn (1.1). Our aim here is to show that there exists a positive time T such that u is positive in Ω 3 ½0; TÞ and u ¼ 0 in Ω 3 ½T; ∞Þ. This T is called the complete extinction time for Eqn (1.1). From the preceding results, the expansion of positivity and a finite time extinction, obtained in [13,15], we find that a nonnegative weak solution of Eqn (1.1) is positive in Ω 3 ½0; T 0 Þ for some T 0 > 0 and u vanishes in Ω 3 ½T Ã ; ∞Þ for some T Ã > T 0 . Here we notice the possibility that a gap T 0 < T Ã may appear and thus, the solution may have positive portion together with zero one in Ω 3 ½T 0 ; T Ã Þ. The proof of the finite time complete extinction is nothing but to show the equality T 0 ¼ T Ã , that is the main issue in this paper. Our main assertion is the following: Theorem 1.1. (Finite time complete extinction). Let n ¼ 3; p ∈ ½2; 3Þ and q þ 1 ¼ 3p 3 − p . Suppose that Ω is a convex bounded domain. Let u 0 be positive, bounded and in W 1;p 0 ðΩÞ. Let u be a nonnegative, continuous weak solution of (1.1). Then there exists a positive T < ∞ such that T is the complete extinction time for (1.1), that is, u is positive in Ω 3 ½0; TÞ and u vanishes in Ω 3 ½T; ∞Þ.
Under the interior positivity and finite time extinction, explained above, our main task is to devise an appropriate comparison function, rely on the comparison theorem and verify that T 0 ¼ T Ã above. We follow the construction of comparison function in [6], where the Laplace operator being p ¼ 2 is studied in any dimensional space domain. Here we shall treat the doubly nonlinear operator, the p-Laplacian coupled with the porous medium operator with the Sobolev critical exponent q þ 1 ¼ 3p 3 − p in three dimensional space domain. We shall compute the p-Laplacain under polar coordinates in the three space dimension, since the higher dimension case seems to be technically difficult. So far, there is no generalized method to evaluate the p-Laplacian in higher dimensional case since this operator is nonlinear and we cannot apply the cylindrical coordinates and the mathematical induction to generalize the case for higher dimension. The convexity of domain is used for the comparison argument to be worked well for our demand (see the proof of Theorem 4.1). Here we also need to assume the continuity of a weak solution to Eqn (1.1), because the regularity for (1.1) is now unknown to be valid in the nonhomogeneous case that q þ 1 is not equal to p, stated as before. In the forthcoming work, we shall study how to remove the assumption of regularity.
The structure of this paper is as follows. In Section 2, we prepare some notations, algebraic inequalities and the definition of weak solution for future use. In Section 3, we gather the fundamental properties of a weak solution of Eqn (1.1) such as the global existence of a weak solution, nonnegativity and boundedness, the so-called expansion of positivity and a finite time extinction of a weak solution. The final Section 4 is devoted to the main theorem and its proof. The proof relies on an appropriate choice of comparison function. Here the computation of the p-Laplace operator is done under the polar-coordinates in three dimensional space domain.
Remark 1.2. We prove our main theorem for the case of critical Sobolev exponent as we have used the Talenti's function which is a unique solution of the stationary equation on all of space ℝ n corresponding to Eqn (1.1) (see [16,17]) to make bound the extinction time.This special function is legitimate for the case of critical Sobolev exponent. Using the usual energy estimates finite extinction time for the subcritical and critical case can be achieved (see [18]). For the supercritical case finite extinction time cannot work well.

Preliminaries
We exhibit in this section some notation, analytic tools, definition and statement of some basic theorems including the comparison theorem used later.

Notation
Following the standard text books [19,20], we set the following notation. Let Ω be a bounded domain in ℝ n ðn ≥ 3Þ with smooth boundary vΩ and for a positive T ≤ ∞ let Ω T :¼ Ω 3 ð0; TÞ be the cylindrical domain. Let us define the parabolic boundary of Ω T by v p Ω T :¼ vΩ 3 ½0; TÞ ∪ Ω 3 ft ¼ 0g: Now, we will present some function spaces, defined on space-time region. For 1 ≤ p; q ≤ ∞; L q ðt 1 ; t 2 ; L p ðΩÞÞ is a function space of measurable real-valued functions on a space-time region Ω 3 ðt 1 ; t 2 Þ with a finite norm kvk L q ðt1;t2; L p ðΩÞÞ :¼ when p ¼ q, we write L p ðΩ 3 ðt 1 ; t 2 ÞÞ ¼ L p ðt 1 ; t 2 ; L p ðΩÞÞ for brevity. For 1 ≤ p < ∞ the Sobolev space W 1;p ðΩÞ consists of measurable real-valued functions that are weakly differentiable and their weak derivatives are p-th integrable on Ω, with the norm . . . ; v xn Þ denotes the gradient of v in a distribution sense, and let W 1;p 0 ðΩÞ be the closure of C ∞ 0 ðΩÞ with respect to the norm k$k W 1;p . Also let L q ðt 1 ; t 2 ; W 1;p 0 ðΩÞÞ denote a function space of measurable real-valued functions on space-time region with a finite norm kvk L q ðt1;t2;W 1;p 0 ðΩÞÞ :¼ The algebraic inequality is often used later on.

Weak solution
Here we are going to define a weak solution which is the most basic prerequisite element to conduct the ongoing research of our Eqn (1.1).
A measurable function u defined on Ω T is called a weak solution to Eqn (1.1) if it is simultaneously a weak sub and supersolution, that is,

Fundamental facts and results
In this section, let n ≥ 3; p ∈ ½2; nÞ and q þ 1 ¼ np n − p , the Sobolev exponent.

Comparison theorem
We here state the comparison theorem being available for Eqn (1.1), that is used in the proof of our main theorem later. The proof is given in Appendix. We say that u ≥ v on v p Ω T in the trace sense, if ðÀ uðtÞ þ vðtÞÞ þ ∈ W 1;p 0 ðΩÞ; for every t ∈ ð0; TÞ; and ðÀ u q ðtÞ þ v q ðtÞÞ þ → 0 in L 1 ðΩÞ as ta0: (3.1) Theorem 3.3. (Comparison theorem, [15]). Let u and v be a weak supersolution and subsolution to (1.1) in Ω T :¼ Ω 3 ð0; TÞ, respectively. If u ≥ v on v p Ω in the sense (3.1), then it holds true that

Expansion of positivity
In this section, we state some positivity results of a nonnegative weak solution to the doubly nonlinear Eqn (1.1), that we will use later to prove our main theorem. We recall that the socalled expansion of positivity holds true for a nonnegative weak solution of Eqn (1.1). This positivity result is proved in [15]. Here we recall them without the proof. The regularity of the solution can be realized upto the positivity region and after that region we do not know about the regularity of the solution of Eqn (1.1).
The following is the expansion of positivity in a compact subdomain. For the proof see [15,Theorem 4.7].
for some k ∈ f0; 1; . . . ; 2 J þI − 1g, where t N is written as If a nonnegative weak supersolution u is positive in Ω at some time t 0 ≥ 0, its positivity may expand in a future time interval starting from t 0 , that is without any waiting time. See the proof in [15,Corollary 4.8].

Extinction of solutions
In this section, we will state the definition of finite extinction time and a proposition which ensures the existence of finite extinction time of a solution to Eqn (1.1). Firstly, the extinction time is defined as follows: (2) uðx; tÞ ¼ 0 for any x ∈ Ω and for all t ≥ T Ã The following proposition presents the finite time extinction of the solution of (1.1).
Proposition 3.8. Let q þ 1 ¼ np n − p , the Sobolev critical exponent. Suppose the initial data u 0 > 0 in Ω: Then there exists T Ã > 0 such that u ¼ 0 in Ω 3 ½T * ; ∞Þ For the proof see Appendix. Here we use the special function peculiar to the Sobolev critical case q þ 1 ¼ np n − p . Proposition 3.9. Let q þ 1 ¼ np n − p , the Sobolev critical exponent. Let u be a nonnegative weak solution to (1.1) in Ω ∞ . Then there exists an extinction time T Ã > 0 for u which is bounded as where Y is Talenti's function defined by with a positive parameter λ.
Talenti's function is an unique solution of the stationary equation on all of space ℝ n corresponding to (1.1) (see [16,17]).

Main theorem
Our main result in this paper is the following theorem.
Theorem 4.1. Suppose that n ¼ 3. Let p ∈ ½2; 3Þ and q þ 1 ¼ 3p 3 − p , the Sobolev critical exponent. Let Ω be a convex bounded domain with smooth boundary. Suppose that the initial data u 0 belong to W 1;p 0 ðΩÞ, positive and bounded in Ω. Let u be a continuous weak solution of Eqn (1.1) in Ω ∞ :¼ Ω 3 ð0; ∞Þ with the initial and boundary data u 0 . Then there exists a positive number T < ∞ such that u > 0 in Ω 3 ½0; TÞ and u ¼ 0 in Ω 3 ½T; ∞Þ.
Proof. By Theorem 3.4 and Proposition 3.8, we have the existence of finite positive T 0 and T Ã such that u > 0 in Ω 3 ½0; T 0 Þ and u ¼ 0 in Ω 3 ½T Ã ; ∞Þ. We notice that the solution u may have a positive portion and a zero one in Ω 3 ½T 0 ; T Ã Þ. Therefore, our aim is to show that T 0 ¼ T Ã . The uniqueness of a weak solution to (1.1) holds true by the comparison principle, Theorem 3.3. Thus, we may assume the following: for any t 0 ∈ ½T 0 ; T Ã Þ, there exists a space point x 0 ∈ Ω such that u(x 0 ; t 0 ) > 0. Indeed, if uðt 0 Þ ¼ 0 in Ω for some t 0 ∈ ½T 0 ; T Ã Þ, then the function u being extended to zero in Ω 3 ½t 0 ; ∞Þ is also a weak solution of Eqn (1.1). That contradicts the choice of t 0 ∈ ½T 0 ; T Ã Þ (see Figures 1 and 2).   where the positive parameters α; β; δ 0 ; m and μ are determined according to the demand, later. As before, we choose α; β and δ 0 so small that R ⊂ Ω 3 ½T 0 ; T Ã Þ. Again, that is possible by the hypothesis that the domain Ω is convex (see . For brevity, we use the abbreviation hereafter  (4.2) We know that The drift term is computed as which is verified by a large positive μ depending on a small β, and the lower positive bound of sin f such that 1 2 Here the reasoning is as follows: since 0 ≤ r ≤ R ¼ diamðΩÞ and t 0 ≤ t ≤ t 0 þ δ 0 , the parameter m can be so small that the quantity in the bracket is positive. Thus, The boundary condition of w is verified as follows: On the lateral boundary, w ¼ 0 because at θ ¼ α; − α; On the arc like boundary of R; w ≤ u if the parameter m is so small that 0 ≤ w ≤ m δ 0 ðdiamðΩÞÞ μ ≤ min A u, where A ¼ B ρ 0 ðx 0 Þ 3 ðt 0 ; t 0 þ δ 0 Þ: On the initial boundary C 3 ft ¼ t 0 g; wðx; t 0 Þ ¼ 0 ≤ uðx; t 0 Þ. Therefore, w is the subsolution of L in R and thus, u ≥ w in R by Theorem 3.3. Hence, the solution u is positive in R: This is true for any R with vertex on the boundary vΩ and thus, u is positive in Ω 3 ðt 0 ; t 0 þ δ 0 Þ, because of the convexity of the domain. Hence the proof is complete. , Note that ∇ðf δ ðv À uÞÞ ¼  Since v t ðv q Þ and v t ðu q Þ belong to L 2 ðΩ T Þ, by the Lebesgue dominated convergence theorem, we can take the limit as δa0 in (A.5) and then obtain, as t 1 a0, Z Ω ðv q ðtÞ À u q ðtÞÞ þ dx ≤ 0 and thus, v q ≤ u q in Ω, for nonnegative t ≤ T, which is equivalent to that vðtÞ ≤ uðtÞ in Ω; for nonnegative t ≤ T. Hence the proof is complete. , Proposition 3.8 is given by Proposition 3.9. Therefore, we are now going to exhibit the proof of the Proposition 3.9.
Proof of Proposition 3.9. The proof is similar to [13]. We consider the solution of the corresponding elliptic equation of (1.1) for the sake of construction of a suitable comparison function and this function is called Talenti function [17] which is defined as where a and b are positive numbers. G. Talenti showed in [17] that this function is the best constant in Sobolev inequality. Furthermore, a straightforward mathematical calculation reveals that Y a;b is a solution of the equation −Δ p Y a;b;z ¼ n n À p p À 1 p−1 a b p−1 Y q a;b;z in ℝ n :

Equation of fast diffusion
In [16], Sciunzi showed that solution of this equation is necessarily of the form Y ðxÞ ¼ Y λ;y ðxÞ :¼ 1 λ n n À p p À 1 To proceed further, we assume by a translation that the origin 0 ∈ Ω. Let u ¼ uðx; tÞ be a nonnegative weak solution to Eqn (1.1). Next, let W ðx; tÞ ¼ X ðxÞTðtÞ be a nonnegative separable solution of v t W q À Δ p W ¼ 0 in ℝ n 3 ð0; ∞Þ: Then X ðxÞ TðtÞ satisfies & ðTðtÞ q Þ 0 ¼ μTðtÞ p−1 in ð0; ∞Þ Δ p X ¼ μX q in ℝ n ; (A.8) where μ is a separation constant. Applying integration by parts to the first separable function, we see that the separation constant μ < 0. Let us set X : A finite-energy solution to (A.9) is given by (A.7). By direct computation, we find that TðtÞ ¼ Tð0Þ where (A.10) is used. The proof is complete. ,

Corresponding author
Md Abu Hanif Sarkar can be contacted at: sarkarhanif@ymail.com For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com