A finite time extinction profile and optimal decay for a fast diffusive doubly nonlinear equation

In this article, we consider a fast diffusive type doubly nonlinear parabolic equation and study the extinction behavior of a solution at a finite time. We show the complete extinction of a weak solution with a nonnegative initial datum, that is, a weak solution is positive before a finite time and vanishes after it, and derive the optimal decay estimates of extinction. Our key ingredient of the proof is a nonlinear intrinsic scaling and the expansion of positivity.


Introduction
We consider the finite time extinction phenomenon for the fast diffusive doubly nonlinear parabolic equations. The doubly nonlinear parabolic equations treated in the paper possess the p-Laplacian coupled with the porous medium operator. Precisely, let Ω ⊂ R n (n ≥ 3) be a bounded domain with the smooth boundary ∂Ω, and let p ∈ (1, n) and q ≥ 1 satisfy p < q + 1 ≤ p * , where p * := np n−p is the Sobolev critical exponent. We shall deal with the following Cauchy-Dirichlet problem for doubly nonlinear parabolic equation: ⎧ ⎪ ⎨ ⎪ ⎩ ∂ t (|u| q−1 u) − Δ p u = 0 in Ω ∞ := Ω × (0, ∞) u = 0 on ∂Ω × (0, ∞) u(·, 0) = u 0 (·) i n Ω . (1.1) Throughout this paper, Δ p u := div |∇u| p−2 ∇u describes the p-Laplacian, where ∇u = (∂ xi u) 1≤i≤n denotes the spatial gradient of u with respect to x, and we assume that the initial datum u 0 belongs to the Sobolev space W 1,p 0 (Ω), and is nonnegative, not identically zero, and bounded in Ω. Our study for the non-homogeneous type doubly nonlinear equation (1.1) is motivated by that of p-Sobolev flow [18,19,25].
In order to formulate our main result, we briefly explain our complete extinction problem for (1.1), the precise notion is presented in Definition 3.4: A positive number t * is called the complete extinction time of (1.1) if a solution u of (1.1) is positive in Ω × (0, t * ), and vanishes in Ω × [t * , ∞).
The complete extinction phenomenon actually holds true for a weak solution of (1.1) with a nonnegative initial data. Our proof employs the expansion of positivity and an intrinsic scaling, that are established in our previous works [18,19,25]. See [9,10] on the positivity for the porous medium and p-Laplace equations. We also extend the expansion of positivity by a stretching transformation of time and a nonlinear scaling method to the subcritical case that p < q + 1 < p * . See [19,Theorem A.6,Proposition 4.6] in the critical case that q + 1 = p * .
The Hölder regularity is well-known to hold for nonnegative weak solutions to the porous medium type equations and the evolutionary p-Laplace equations; for instance see [8,29]. The local regularity for doubly nonlinear parabolic equations also have been studied by Vespri [37,38], Porzio and Vespri [26], and Ivanov [15,16], where the case that p ≤ q+1 is treated, but the class of weak solutions is somehow different from ours. See further references [12]. The proofs of the regularity for general doubly nonlinear parabolic equations are based on De Giorgi's alternative approach with the intrinsic scaling method, originally introduced by DiBenedetto (Fig. 1). For the so-called Trudinger's equation in the homogeneous case that q + 1 = p, the Harnack inequality and local regularity for nonnegative weak solutions are proved by Kinnunen and Kuusi [17] et al. There also exists a viscosity approach for the doubly nonlinear equation by Bhattacharya and Marazzi ( [4]).
In the present paper, one of our main theorems is below. Theorem 1.1. (Finite complete extinction of (1.1)) Let p ∈ (1, n) and q ≥ 1 be such that p < q + 1 ≤ p * = np n−p . Suppose that the initial datum u 0 belongs to W 1,p 0 (Ω), and is nonnegative, not identically zero, and bounded in Ω. Let u be a weak solution to (1.1) in the sense of Definition 3.1. Then there is a complete extinction time t * of u, that is, u is positive in Ω × (0, t * ) and u vanishes in Ω × [t * , ∞). Moreover, the solution u and its gradient are locally Hölder continuous in Ω × (0, t * ).
Our second result is the extinction profile at the finite complete extinction time in the following: Theorem 1.2. (Asymptotic convergence) Let p ∈ (1, n) and q ≥ 1 be such that p < q + 1 ≤ p * = np n−p . Suppose that the initial datum u 0 ∈ W 1,p 0 (Ω) is bounded, nonnegative and not identically zero. Let u be a weak solution to (1.1) in the sense of Definition 3.1 and t * be the extinction time of u in the sense of Definition 3.4. Then the following statement holds: NoDEA A finite time extinction profile Page 3 of 48 43 x y Ω u 0 0, 0 Figure 1. Phenomenon of the complete extinction • In the subcritical case p < q+1 < p * , for any increasing sequence t k t * , there exist a subsequence {t k } with the same notation, and a nonnegative function U ∈ W 1,p 0 (Ω)\{0} such that as k → ∞, where U is a weak solution, which is not identically zero but a nonnegative, to the Dirichlet problem with a constant λ p,q = q/(q + 1 − p). • In the critical case q + 1 = p * , the weak convergence in W 1,p 0 (Ω) to a nonnegative solution of (1.3) holds true for some subsequence of any increasing time-sequence t k t * .

Remark 1.3.
Let us comment on the critical case q + 1 = p * for Theorem 1.2.
(a) In the critical case q + 1 = p * , there is only shown to hold the weak convergence in W 1,p 0 (Ω) and its weak limit can be trivial; therefore the energy and volume gap at the weak limit U may appear in at most finitely many points in Ω by the energy and volume boundedness (see the end of proof of Theorem 1.2). Then, we may have the so-called energy and volume concentration at the limit (for p = 2 see [5,34] and references therein). Moreover, the concentration of energy and volume is given as the limit of scaled solutions on space, where the scaling transformation makes invariant the elliptic parts of the evolution equation for the left-hand side of (1.2) (see (5.2)), which is the same as stationary equation (1.3) 1 . This scaling limit may be regarded as a microscopic limit of the left-hand side of (1.2). Therefore, the weak limit at infinity time is characterized as the 43 Page 4 of 48 M. Misawa, K. Nakamura and Md. A. H. Sarkar NoDEA sum of the macroscopic limit at finitely many concentration points and the reminder. The reminder is the weak limit of solutions on the left-hand side of (1.2). The proof of the phenomenon expected in the critical case q + 1 = p * is based on the local boundedness of solutions on the lefthand side of (1.2) depending on the situation whether the local volume is uniformly bounded or not. This will be revealed in our forthcoming paper. The strong maximum principle for the positivity of scaled limit is in the literature, cf. [36]. If the convergence is strong as in (1.2), that is, no energy (volume) gap does not appear, the limit is a positive solution of (1.3). The existence of a positive solution of (1.3) may depend on the geometry of domain (refer to [30,32] for R n and a compact manifold, and also see [ As for the finite time extinction phenomenon of the plasma equations or the porous medium equations, those are given (1.1) with p = 2, a by now large literature is available; for instance, it is well known that this problem is originally addressed by Berryman and Holland [3]. After that, assuming the regularity for nonlinear term that mediates solution itself and geometric condition on the domain, ) established the finite time complete extinction of a continuous weak solution in terms of appropriate comparison function in any dimension, where the continuity of a weak solution of the porous medium equation is essentially used (see the regularity in [8,29]). See also [2,7] for details. In the Laplacian and 1D case, Sabinina proved a finite time extinction of the plasma equation [28]. Savaré and Vespri showed the asymptotic profile of doubly nonlinear parabolic equations in the case that q + 1 < p * (see [31,Remark 4.6]), where the equation is formally equivalent to (1.1) by a changing of unknown function v = |u| q−1 u. This procedure enables to avoid the non-linearlity in the time-derivative and work in the L 2 -framework. In contrast, we shall tackle the power nonlinearity in the time derivative in a direct fashion, which is a one of our motive. The large time behavior of solutions to Eq. (1.1) in the case q < 1 is established by Stan and Vázquez [33]. However, the solution class is different from ours. For the p-Laplacian in 3D case and the critical case q + 1 = p * , the complete extinction of a continuous weak solution to (1.1) in a convex domain is shown via constructing appropriate comparison functions [27]. The optimal decay estimate and stability of asymptotic profile in the Laplacian case p = 2 is studied by Akagi and Kajikiya [1], where the solution class is different from ours. Recently, in the sensational paper [5] by Bonforte and Figalli, the sharp extinction rates are established for (1.1) with NoDEA A finite time extinction profile Page 5 of 48 43 p = 2, whose approach is based on the so-called Nonlinear Entropy Method, combined with the spectrum analysis of the Dirichlet Laplacian in weighted L 2 -spaces. We emphasize that Theorem 1.1 contains the results for the plasma or porous medium equations and the proof does not require any continuity of a weak solution, in contrast to those of the above results. Our approach based on the expansion of positivity may be of its own interest. The decay estimates rely on some energy estimates, obtained from an appropriate approximation of the equation (1.1). The approximation yields the rigorous derivation of energy estimates, that is demonstrated in Appendix B. The energy estimates lead to a monotonicity of the so-called Rayleigh quotient, that is the key to the decay estimates. See Lemma 4.1 and Appendix D below (Also refer to [1,20,21]).

Organization of the paper
The outline of the paper is in the following. In the next section, we give some notation and recall some fundamental tools used later. Section 3 is devoted to the global existence and regularity estimates for the doubly nonlinear parabolic equation (1.1). We further present the nonlinear intrinsic scaling, that transforms the prototype doubly nonlinear equations to the p-Sobolev type flow, and give the proof of our main result, Theorem 1.1. In Sect. 4, we derive the optimal decay estimate for Eq. (1.1). In Sect. 5, we prove our second result, Theorem 1.2. In Appendix A we give the uniqueness of nonnegative solutions to (1.1) and its transformed equation. Appendix B is devoted to construction of an approximating solution to (1.1) satisfying the energy estimate as in Proposition B.9. The quantitative estimates and convergence result for approximate equation (B.3) are given in Appendix C. In Appendices D and E, we collect the proof postponed in the previous section.

Notation
In the following, we fix some notation which will be used throughout the paper. Let Ω ⊂ R n (n ≥ 3) be a bounded domain with smooth boundary ∂Ω. For a positive T ≤ ∞, let Ω T := Ω × (0, T ) be the space-time domain.
From now on we denote by C, C 1 , C 2 , · · · different positive constants in a given context. Relevant dependencies on parameters will be emphasized using parentheses. For instance C = C(n, p, Ω, · · · ) means that C depends on n, p, Ω · · · . Further, a general positive constant C will not necessarily be the same at different occurrences and can also change from line to line. For the sake of readability, the dependencies of the constants will be often omitted within the chains of estimates. In addition, the n-th line of the Eq. ( · ) is denoted by the symbol ( · ) n . With S ⊂ R k being a finite measurable set with Lebesgue measure |S| and with f being an integrable function on S, we shall denote the integral average by We also abbreviate the essential infimum and essential supremum as ess inf ≡ inf and ess sup ≡ sup, respectively.

Function spaces
We recall some function spaces used throughout the paper. Let 1 ≤ p, q ≤ ∞. For a Banach space X we use the space of Bochner L q (t 1 , To avoid confusion, we shall deal with the above Lebesgue spaces except p = q = ∞. Choosing X as the Sobolev space on Ω, W 1,p 0 (Ω), we have the space Again, we will omit to consider the case p = q = ∞. In addition, for an interval I ⊂ R, by C(I; L q (Ω)) we denote the space of all continuous functions I t → u(t) ∈ L q (Ω).

Fundamental tools
We shall present the fundamental tools often used.
Let us define the Rayleigh quotient by which is involved the well-known Sobolev-Poincaré inequality provided q + 1 ≤ p * . Notice that the best constant C p,q is achieved by We record the following algebraic inequality throughout the paper.
where the symbol · denotes the inner product on R k . In particular, when p ≥ 2 Proof.

Fundamental properties of a weak solution; Proof of Theorem 1.1
In this section, we collect some results required for the proof of Theorem 1.1.
We can refer to the proofs of results in [18,19,23,25] (also see [24]). We first present some fundamental properties of a weak solution to (1.1) and then, we prove Theorem 1.1. Recall that 1 < p < n and q ≥ 1 satisfy p < q + 1 ≤ p * = np n−p .
To begin, we give the definition of weak solutions of (1.1).
Definition 3.1. (Weak solutions)Let u = u(x, t) be a measurable function defined on Ω ∞ := Ω × (0, ∞). We call u as a weak supersolution (subsolution) of (1.1) in Ω ∞ provided that the following conditions (i)-(iii) are satisfied: (iii) u attains the initial data continuously in the Sobolev space: and satisfies the boundary condition in the trace sense: We call a measurable function u defined on Ω ∞ as a weak solution to (1.1) if u is simultaneously a weak super and subsolution. Theorem 3.2. (Global existence of (1.1) cf. [18,25]) Let p ∈ (1, n) and q ≥ 1 satisfy p < q + 1 ≤ p * . Suppose that the initial value u 0 is in the Sobolev space W 1,p 0 (Ω), nonnegative and bounded in Ω. Then there is a global in time weak solution u of (1.1) in the sense of Definition 3.1, which is nonnegative and bounded in Ω ∞ , that is, (3.1) Additionally, the energy identity holds whenever t 1 , t 2 ∈ [0, ∞) with t 1 < t 2 and, the following integral inequalities hold true for every t > 0: with C ≡ C(n, p) being a positive constant and u(t) p := u(t) L p (Ω) being shorthand notation.
Proof. The proof can be achieved similarly as in [24] and also [18,25]; therefore, we shall only sketch it. Note that the arguments in [18,25] are applicable to the both critical and subcritical cases p < q + 1 ≤ p * with 1 < p < n and q ≥ 1. Equation (  This is deduced by the construction of the approximate solutions or the exponential mollification method and the result can be extended to all q > 0. The precise description is addressed in [24]. By the nonnegativity of the solution u to (3.1), we hereafter substitute the following equation (3.6) for (1.1): in Ω.
We now recall the definition of the extinction time of a weak solution u to (3.6) (see [19]).

Definition 3.4. (Extinction time)
Let u be a nonnegative weak solution to (3.6) in Ω ∞ in the sense of Definition 3.1. We call a positive number t * the extinction time of u if it satisfies (i) u(x, t) is nonnegative and not identically zero on Ω × (0, t * ) (ii) u(x, t) = 0 for any x ∈ Ω and all t ≥ t * .
The finite time extinction for (3.6) actually holds true, as stated below.
NoDEA A finite time extinction profile Page 9 of 48 43 Proposition 3.5. (Finite time extinction for (3.6)) Let 1 < p < n and q ≥ 1 satisfy p < q + 1 ≤ p * . Let u be a nonnegative weak solution to (3.6) in Ω ∞ in the sense of Definition 3.1. Then there is a extinction time t * > 0 of u in the sense of Definition 3.4, which is bounded from above as follows: , where λ p,q := q/(q + 1 − p) and C p,q is the best constant as in the Sobolev-Poincaré inequality (2.2).
Proof. The proof simply follows from (3.2) in Theorem 3.2 and the Sobolev-Poincaré inequality (2.2). The full proof can be seen in [23,Proposition 3.4].
We shall present the following nonlinear intrinsic scaling (see [19, Proposition 4.1] and also [23,Proposition 4.1]), which is the key ingredient to derive the strict positivity of the solution u before the extinction time.
Proposition 3.6. (Nonlinear intrinsic scaling) Let u be a nonnegative weak solution to the equation (3.6) in Ω ∞ and let t * < +∞ be a finite extinction time of u. There exist unique and, subsequently, such that the following is valid: Let and set (3.10) Then w is a nonnegative weak solution of the doubly nonlinear parabolic equa- |∇w(x, τ )| p dx and the initial value w 0 is defined as u 0 / u 0 q+1 with u 0 being the initial data as in (3.6). The definition of a Theorem 3.7. (Strict positivity for solutions w of (3.11)) Let w be a nonnegative weak solution to (3.11) with initial data w 0 = u 0 / u 0 q+1 , defined by (3.10), where the original initial data u 0 is as in (3.6). Then there holds true that We are now in position to prove Theorem 1.1.
Proof of Theorem 1.1. We will prove Theorem 1.1 separately as follows.
Step 1 (Complete extinction) From (3.12) in Theorem 3.7 it follows that for every t 0 < t * because γ(t) > 0 for any nonnegative t < t * in the definition (3.10). Hence the complete extinction of solution u is actually verified.
Step 2 (Regularity of solution w to (3.11)) At first we derive the boundedness of a weak solution w to (3.11), where we use the fact that λ(τ ) = ∇w(τ ) p p in (3.11). See the proof in [18, Propositions 3.5 and 5.3]. Proposition 3.8. (Boundedness of (3.11)) Let w be a nonnegative weak solution of (3.11) in Ω T for any positive T < ∞. Then w is bounded from above in Ω T and Following the exactly similar argument to [18, Sect. 5.2], we can deduce the Hölder and the spatial gradient Hölder regularity of solution w to (3.11) with initial datum w 0 = u 0 / u 0 q+1 (also see [19,Sect. 5] and [23, Theorem 6.4]).
Let T ∈ (0, ∞) and Ω be any subdomain compactly contained in Ω. As is noticed in Theorem 3.5, we have the expansion of positivity for the solution w, that together with Proposition 3.8 yields the bounds from above and below of a solution w to (3.11) NoDEA A finite time extinction profile Page 11 of 48 43 for some m > 0. Equation (3.11) 1 can be written as follows: and hence, W is a positive and bounded weak solution of the evolutionary p-Laplacian equation (3.14) in Ω T . By (3.13) G is actually uniformly elliptic and bounded in Ω T . The right-hand side of (3.14) is bounded, which is assured by (3.13), The following Hölder continuity is verified by the local energy inequality for a local weak solution W to (3.14) ( [18, Lemma C.1]) and standard iterative real analysis methods. See also [9, Chapter III] more details. Theorem 3.9. (Hölder continuity) Let W be a positive and bounded weak solution to (3.14) and let Ω Ω be a subdomain. Then W is locally Hölder continuous in Ω × (0, T ] with a Hölder exponent β ∈ (0, 1) on the parabolic metric |x| + |τ | 1/p .
By the positivity and boundedness (3.13) and the Hölder continuity in Theorem 3.9, the coefficient function G p−1 is lower, upper bounded and Hölder continuous and thus, a Hölder continuity of the spatial gradient holds true. By using an elementary algebraic estimate and a interior positivity, boundedness, the Hölder regularity of W and its gradient ∇W in Theorems 3.9 and 3.10, we can bring out the local Hölder regularity of the weak solution w to (3.11) and its gradient ∇w. Step 3 (Regularity of original solution u to (3.6)) Finally, the local Hölder and gradient local Hölder regularity of the original solution u to (3.6) follow from Theorem 3.11 and the definition of w in (3.10), where γ(t) in (3.10) is continuous for any nonnegative t < ∞ by (3.2) in Theorem 3.2 and the regularity of transformations in (3.10) with (3.7)-(3.9).
Theorem 3.12. (Hölder continuity) Let u be a nonnegative and bounded weak solution to (3.6) and t * be the extinction time of u. Then the following implications hold: an exponent β ∈ (0, 1) on the usual parabolic metric |x| + |t| 1/2 .
Therefore, the proof of Theorem 1.1 is concluded.

Decay estimates
In this section, we shall derive the decay estimates for a nonnegative weak solution of (1.1).

Lemma 4.1.
Let u be a nonnegative weak solution to (3.6). Let t * be an extinction time of u. Then for any t ∈ [0, t * ) This lemma is shown by an approximating argument as in [23,Sect. 7], and the proof will be given in Appendix D.
The main result of this section is condensed in the optimal decay estimates. Proposition 4.2. (Estimates from above and below) Let 1 < p < n and q ≥ 1 satisfy p < q + 1 ≤ p * . Let u be a nonnegative weak solution to (3.6) and t * be the extinction time of u. Then, for every nonnegative t ≤ t * , the following estimations hold true: Proof. We shall shorten Φ(t) := u(t) q+1 q+1 . Multiplying (3.6) by u and integration-by-parts render us with denoting := d dt . By Lemma 4.1 R(u(t)) is bounded from above by R(u 0 ) and it readily follows from (4.4) that . Thus, the first assertion (4.1) is actually verified. Again, by (4.4) and the Sobolev-Poincaré inequality (2.2) we gain Integrating this over (0, t * ) again yields which is the desired result (4.2).
Finally, we now turn to prove (4.3). Using the definition of Rayleigh quotient and (4.1), we get Since by Proposition 4.1 R(u(t)) is bounded from above by R(u 0 ), the above inequality leads to (4.5) By use of (4.2) and the Sobolev-Poincaré inequality (2.2) again, we also bound which together with (4.5) in turn implies the last inequality (4.3) and therefore the proof is complete. Proposition 4.2 yields the optimal extinction rate and the estimations of the extinction time t * ≡ t * (u 0 ) from above and below.

Corollary 4.3. (Extinction rate and extinction time)
Let u be a nonnegative weak solution of (3.6). Then the optimal extinction rate of u is given by (4.7) Furthermore, the extinction time t * ≡ t * (u 0 ), depending on the initial datum u 0 , is estimated as Proof. By Proposition 4.2, the optimal extinction rate is given by By (4.1) and (4.2) in Proposition 4.2, it holds that Passing to the limit as t 0 in the display above, the assertion is readily follows.

Asymptotic profile; Proof of Theorem 1.2
In this section we prove the asymptotic convergence as stated in Theorem 1.2.

Transformation stretching the time-interval
We introduce a transformation stretching time-interval, that extends timeinterval up to the extinction time into infinite one. The extinction profile is clearly determined by the stationary problem associated with the transformed evolution equation (Fig. 2). The method is nowadays well known for singular parabolic equations (refer to [20,21,31]). Let t * be an extinction time of a nonnegative weak solution u to (3.6). We define, for any By simple manipulation, v = v(x, s) solves the following equation where λ p,q := q/(q+1−p) is a fixed constant. More precisely, v is a nonnegative weak solution to (5.2) in the following sense.   .2) in Ω ∞ provided that the following conditions are satisfied: and v is bounded from above by Formally, the direct calculation shows that Combining the preceding formulae, we have that ∂ s t = t * − t and It is certain by (5.3) that v ∈ L ∞ (0, ∞ ; L q+1 (Ω)). Hence, for given positive S < ∞, setting T := t * 1 − e −S and using (3.1), (3.5) and Hölder's inequality, we gain In change of variable s = log in mind, we observe that, for every testing function  for every s 0 = log t * t * − t 0 < +∞ and any positive t 0 < t * with t * being the extinction time of u.
In the final step, we show the sketch of the proof of (5.5). For 0 ≤ s 1 < s 2 < ∞ and δ > 0 small enough, we define the following Lipschitz cut-off function: We further define, for any δ > 0 We choose a testing function as e −λp,qs χ δ (s)ϕ δ (v) in the weak formulation (D2) and take the limit as δ 0 similarly as [ as desired. Therefore the proof is complete.
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. The proof of theorem 1.2 now goes in several steps.
Step 1: For any increasing sequence 0 where N is a negligible subset in (0, ∞) with respect to Lebesgue measure on R. By means of Hölder's inequality, (5.6) and in the limit k → ∞, which validates (5.8). With this θ k , by letting t k : as k → ∞, and therefore, via (D2) in Definition 5.1 it holds that for every ϕ = ϕ(x) ∈ C ∞ 0 (Ω). Since by (5.3) and (5.4) {v(θ k )} is bounded in W 1,p 0 (Ω), there exist a (non-relabeled) subsequence {θ k } and a limit function U ∈ W 1,p 0 (Ω) so that v(·, θ k ) → U weakly in W 1,p 0 (Ω) (5.10) in the limit k → ∞, where we used Mazur's theorem implying that the closed subspace W 1,p 0 (Ω) of W 1,p (Ω) is weakly closed in W 1,p (Ω). Moreover, this together with the compact embedding W 1,p 0 (Ω) → L r (Ω) for all r ∈ [1, p * ) yields the strong convergence v(·, θ k ) → U strongly in L r (Ω), ∀r ∈ [1, p * ), (5.11) therefore, up to extract a (non-relabeled) subsequence, we deduce that v(·, θ k ) → U a.e. in Ω. (5.12) Step 2: In this step, we are going to show the above weak limit U is actually nonnegative weak solution to (1.3). Fort this, we first deduce the strong convergence of the gradient in order to derive the convergence of the weak form of p-Laplacian and lower order terms appearing in (5.9). where U is the limit function as in the procedure leading to (5.10).
The proof of this lemma is postponed, and will be given in Appendix E. The Hölder inequality, the algebraic inequality (2.3), the estimate (5.4) and the strong convergence of the gradient, Lemma 5.3 give the estimations:
In the subcritical case q + 1 < p * , we shall prove the strong convergence of v(θ k ) to U in W 1,p 0 (Ω). For this we first prove that as k → ∞. In fact, since by q ≥ 1 the chain rule of weak differential implies

Then, by means of Hölder's inequality and (5.3), it is
which converges to zero, since ∂ s v (q+1)/2 (θ k ) 2 → 0 in the limit k → ∞. At this stage, we now subtract (5.15) from (5.9) and use the test function ϕ = v(θ k ) − U in the resulting equation. Similarly as the manipulation leading to (5.13) and (5.14), by the algebraic inequality (2.5) if p ≥ 2 then as k → ∞, where in the final line we used (5.16) and the strong convergence of v(θ k ) to U in (5.11). In the remaining case 1 < p < 2, by the Hölder inequality and (2.4) we see that as k → ∞, where in the last line we used (5.4) and the strong convergence of v(θ k ) to U in (5.11) again. In the subcritical case q + 1 < p * , the solution U to (1.3) obtained as the limit of v(θ k ) also inherits the same boundedness as in (5.3) and (5.4) from the solutions v(θ k ) by virtue of the strong convergence of v(θ k ) to U in W 1,p 0 (Ω).

By (5.3) and (5.4) for U , the solution U is not identically zero and nonnegative in Ω.
On the other hand, in the critical case q +1 = p * , the solution U does not satisfy the lower boundedness in (5.3) and (5.4) because of the restriction of convergences (5.10) and (5.11). Thus, as well-known, the energy and volume gap may appear along the limitation of v(θ k ) to U as k → ∞.
This completes the proof of Theorem 1.2.
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Appendix A: Uniqueness of a weak solution to (3.6) and (5.2)
Now we shall state the uniqueness of a weak solution to (3.6) and (5.2).
The proof is based on the comparison theorem, retrieved from [18, Theorem 3.6]. Here it is decisive that the weak time-derivative of powered solution in the equation is integrable in space-time.

Appendix B: Approximation of the doubly nonlinear equation (3.6)
This appendix is devoted to regularity estimates for an approximation solution of the doubly nonlinear equation (3.6), which is the key to deriving energy estimates for the original solution u to (3.6), as stated in Proposition B.9 in Sect. 5.1 below. Following [23, Sect. 7.2], we proceed argument as follows.
Let ε, δ be arbitrary positive numbers such that ε 0, δ 0 later. We shall show the existence and some regularity estimates for nonnegative solutions u = u ε,δ to the equation where the initial value u 0 is as in (3.6), which in turn becomes, by setting The straightforward computation shows that V ε solves with the constant λ p,q = q q+1−p .
and u satisfies the integral estimates, for all t ≥ 0, finishing the proof.

B.2. Regularity of solutions to Eq. (B.1)
For the reader's convenience, following [23, Sect. 7.2], we collect the regularity results for Eq. (B.1), which enable us to deduce the energy inequality, as stated in Lemma B.9. Let u be a weak solution of (B.1). Set μ := (u + ε) q ⇐⇒ u = μ 1 q − ε. This procedure transforms (B.1) into the following problem Due to (B.4) in Proposition B.1 there holds that Set μ − ε q = ν, then (B.6) becomes The condition (B.7) is written for the solution ν to (B.8) as The boundary regularity can be transformed into that in the interior case in the usual way (refer to [9, Chapter X, Sect. 1-2, pp. 292-296] and also [11, Appendix C.1, Page 710]). Take an arbitrary point (x 0 , t 0 ) ∈ ∂Ω × (0, T ) and introduce a change of coordinates which maps a small portion of the boundary ∂Ω around (x 0 , t 0 ) into a portion on hyperplane. We consider the equation (B.8) in the local parabolic cylinder where the positive number R satisfies R < t 0 . If necessary, the radius R is chosen small and the portion of the boundary ∂Ω ∩ B R (x 0 ) can be represented as the graph of a smooth function γ : R n−1 → R. Let us define the following change of variables and assume that ∇xγ(x 0 ) = 0; therefore ∇ȳγ(ȳ 0 ) = 0, by letting the graph of γ at x 0 tangent to the (n − 1)-dimensionalx-plane. We shall shorten y = Φ(x). we denote the transformation with the same notation as the original ν in (B.8).
Moreover, we set and, the coefficient matrix (a ,k (y)) is given by Hereafter, the summation convention on repeated indices is used and the dependence on approximation parameters ε and δ of the function f will be kept in mind.
Here we notice the uniformly ellipticity and boundedness condition for the coefficient (a k, (y)): For any η = (η, η n ) ∈ R n where we used Cauchy's inequality and the size R of the local parabolic cylinder can be chosen so that |∇ȳγ(ȳ)| is sufficient small for anyȳ ∈ B R (y 0 )∩{y n = 0}, since ∇ȳγ(ȳ 0 ) = 0 and γ(ȳ) is smooth. Note that the solution ν to (B.8) is nonnegative and we may treat (B.10) 1 as With the notationȳ = (y 1 , . . . , y n−1 ),ν denotes the odd extension of ν in the cylinder Q R (z 0 ), that is, (B.14) and therefore, by the reflexion principle,ν solves Under this setting, by (B.9) which in turn implies that, for any η = (η 1 , . . . , η n ) ∈ R n , if p ≥ 2, then where in the second and penultimate lines we used condition (B.12). If 1 < p < 2, then it holds that the inequalities with the upper and lower bounds replaced each other in (B.17). We will collect the regularity of the solutionν.
and thus, by (B.16) ∇ 2 yν is also squared-integrable locally in Q R (z 0 ). Proof. The assertion is proved by the difference quotient method, whose detail is found in the proof of [23,Proposition 7.5] (also refer to [ a k , ∇y k ν∇y ν ⎞ ⎟ ⎟ ⎠ and set the coefficient matrices of the lower order terms as We have the uniform ellipticity and boundedness for the coefficient (A ,k ): and, whenever 1 < p < 2, where

B.3. Energy estimates for solutions to (B.1)
Here we derive the energy estimates for weak solutions to (B.1). This energy estimates will play a central role in the weak compactness theory for constructing a weak solution to the limiting equations (3.6) of (B.1) as ε 0 and δ 0. Firstly, we establish the energy identities for weak solutions to (B.1), whose proof can been seen in [23,Lemma 7.8].
Lemma B.7. (Energy identities for solutions to (B.1)) Let u = u ε,δ be a weak solution to (B.1). Then, for any nonnegative t 1 < t 2 < ∞, the following energy identities are valid.
Proof. As shown in Lemma B.8, u ∞ is the weak solution to (3.6) and thus, multiplying Proof. The uniqueness for a weak solution to (3.6) follows from Theorem A.2 with considering λ p,q = 0. Thus, the energy estimates hold true for the weak solution to (3.6) by Proposition B.9.

Appendix C: Quantitative estimates for the transformed doubly nonlinear equation (B.3)
In this appendix, using the higher regularity for u as in Lemma B.7, we shall derive the energy inequality for a solution of (B.3).