Existence and nonexistence of global solutions for doubly nonlinear diffusion equations with logarithmic nonlinearity

In this paper, we study an initial-boundary value problem for a doubly nonlinear diffusion equation with logarithmic nonlinearity. By using the potential well method, we give some threshold results on existence or nonexistence of global weak solutions in the case of initial data with energy less than or equal to potential well depth. In addition, the asymptotic behavior of solutions is also discussed.


Introduction
In this paper, we will study the following doubly nonlinear diffusion equations with logarithmic nonlinearity where Ω ⊂ R n (n ≥ 1) is an open bounded domain with smooth boundary ∂Ω, u (m−1) := |u| m−2 u, ∆ p (u) := div (∇u) (p−1) the usual p-Laplacian operators and f q is of the form of logarithmic term f q (s) = s (q−1) log |s|.
Let us consider the following equation which is so-called doubly nonlinear parabolic equations Corresponding author.Email: lxuantruong@gmail.comwhere f (u) is a source if f (u) ≥ 0, whereas f (u) is called a sink.This equation generalizes many equations such as heat equation (as m = 2 and p = 2), the porous medium equation (as m > 1 and p = 2), and the p-Laplacian equation (as m = 2 and p > 1).The equation (1.2) can be divided into three following cases which are called the degenerate, critical and singular case, respectively.
Regarding of the global existence and nonexistence results, there are some well-known methods to study the equation (1.2) depends on whether Ω is bounded or unbounded domain in R n .For example, in the case Ω = R n , Fujita [9] studied the initial value problem for heat equations with power nonlinearity f (u) = u p and then Levine in the survey [18] has extended the results of Fujita for more general parabolic equations with nonlinear dissipative terms in unbounded domains.In [25], Pokhozaev and Mitidieri introduced the nonlinear capacity method in order to study of questions on the blow up of solutions of many nonlinear partial differential equations and inequalities.It is noting that although these methods are really powerful tools to treat the case of nonnegative nonlinearities f (u), it cannot be applied to the case of sign-changing nonlinear terms.And therefore, this method cannot be applied to our problem.
On the other hand, in the case of bounded subdomain of R n , we refer the seminar papers of Kaplan [16], Levine [17] and Ball [3] in which the authors proved the blow up results under condition of non-positive initial energy.In [27], Payne and Sattinger developed the potential well method which is introduced by Lions [20] and Sattinger [30] to study the existence and nonexistence of global weak solutions to heat and wave equations with power like nonlinearity under condition of positive initial energy.More precisely, in [27] the authors show that if the initial energy J(u 0 ) < d, then weak solution u(t) to equation (1.2) (for m = p = 2) is global provided that u 0 ∈ W (stable sets) and blows up in finite time provided that u 0 ∈ U (unstable sets).Afterward, analogous results have been studied extensively to various kind of equations.We refer to [6,[10][11][12][13]17,21,33] for many heat and wave equations and [7,19] for porous medium equations.
In the case of p-Laplacian equation, Tsustumi [31], Fujii [8] and Ishii [15] studied the initial-boundary value problem for the equation where f (u) = |u| q−2 u, with 2 ≤ q < p * = np n−p .As for the existence and nonexistence of global weak solutions to (1.5), the following results are well-known: (i) if p > q, (1.5) admits a global weak solution for any u 0 ∈ W 1,p 0 (Ω); (ii) if p < q, then weak solution u(t) of (1.5) is global when initial data u 0 ∈ W 1,p 0 (Ω) is in stable sets and blows up in finite time when u 0 ∈ W 1,p 0 (Ω) is in unstable sets.
(iii) when p = q, Fujii [8] derived sufficient conditions on blow-up of solutions depending on first eigenvalue λ 1 of the operator −∆ p .
Although there is a lot of results of global existence and nonexistence of weak solutions to (1.2) in the case of power nonlinearities and its generalization, there is a little known about the one with logarithmic nonlinearity.We refer to [4,5,26] for a few recent papers involving logarithmic nonlinearity.In this paper, in the same spirit with previous works, we utilize the potential well method to study the existence and nonexistence of global weak solutions to (1.2) with logarithmic nonlinearities f q (u) = (u) (q−1) log |u|, q > 2 and initial value u (m−1) 0 belonging to Sobolev space W 1,p 0 (Ω).Roughly speaking, our results are as follows: 1) admits a global solution for each u then there exists a weak solution to (1.1) which is global provided that u 0 belonging to stable sets, and blows up provided that u 0 belonging to unstable sets.In addition, decay estimates are also proved for the former case.
where m > 1 is Hölder conjugate of m satisfying 1 m + 1 m = 1, then one has ϕ u (m−1) = u.Hence, by changing variable w = u (m−1) , the equation (1.1) leads to the reformulated equation It is also noticed that f γ (s) is nonhomogeneous and can change signs for s ∈ (0, +∞).In addition, since lim s→0 + f γ (s) = 0, it can be extended continuously to the function fγ with fγ (0 In what follows, for the sake of brevity, we still denote fγ by f γ with noticing that f γ (0) = f γ (1) = 0.The nonlinearity with such properties can be found in the paper [2].Hence, instead of (1.1) we consider the following initial boundary value problems where u 0 ∈ W 1,p 0 (Ω) and γ = (m − 1) (q − 1) + 1 > 1, m is Hölder conjugate of m.The rest of this paper is organized as follows: Section 2 devotes to preliminaries in which we establish some properties of stationary problem associated to (1.7) and introduce the stable sets (potential well) and unstable sets as well as its properties; Section 3 states main results of this paper and their proofs are presented in the remaining sections.

Local minima and potential wells
In this section, we need the following logarithmic Gagliardo-Nirenberg inequality which was introduced by Merker [24].Lemma 2.1 ([24], Logarithmic Gagliardo-Nirenberg inequalities).The inequalities |u| q u q q log |u| q u q q dx ≤ 1 1 − q/p * log C q n,p,q ∇u q p u q q (2.1) Hereby the constant C depends on n and p only in the case p < n, and on n, p and a finite upper bound of q in the case p ≥ n.
This inequality can be reformulated in parametric form.Here, one introduces the following parametric form of logarithmic Gagliardo-Nirenberg inequalities for all µ > 0 where 0 < r ≤ min{p, q}.By virtue of Young's inequality, one obtains the following proposition.
Proposition 2.2 (Parametric form of logarithmic Gagliardo-Nirenberg inequality).Let us suppose all assumptions in Lemma 2.1.Then we have |u| q log |u| q u q q dx + C r n,p,q,µ u q q ≤ µ r p ∇u p p + µ p − r p u (q−r) p−r p q , for all µ > 0 where 0 < r ≤ min{p, q} and C r n,p,q,µ is a constant given by C r n,p,q,µ = qp * (p * − q) r log (p * − q)rµe qp * C r n,p,q .
For u ∈ W 1,p 0 (Ω), we can define u(x) = 0 for x ∈ R n \Ω.Then u ∈ W We now define the energy functional J and Nehari functional I on W 1,p 0 (Ω) related to the problem (1.7) as follows It is clear that the functionals I and J are continuous on W 1,p 0 (Ω) and We also define the Nehari manifold as follows We shall see below (see Lemma 2.5) that each half line starting from the origin of the phase space W 1,p 0 (Ω) intersects the Nehari manifold N exactly once.It is also useful to understand the Nehari manifold N in terms of the critical points of the fibrering map λ → J (λu) for λ > 0 given by Then we have This implies the following lemma immediately.
We now define the depth of potential well which is also characterized as Proof.The case γ = p can be proved similarly to [26].It remains to consider the case γ > p.
Let u ∈ N , then it follows by (2.5) On the other hand, by logarithmic Gagliardo-Nirenberg inequality, one has where r ∈ (0, p) is a constant and C r n,p,q,µ is a constant given by Proposition 2.2.By choosing µ = γp (m −1)r then we get It is noticed that for r ∈ (0, p) and p < γ then γ−r p−r p > γ.And therefore, we deduce from (2.10) that there exists a positive constant R independent of u such that u γ ≥ R > 0 which implies Here S p,γ stands for the best constant in the Sobolev embedding W Thus, the proof follows from (2.9) and (2.11).
Denote the nontrivial stationary solution of problem (1.7) by Then, by virtue of critical point theory, it is not difficult to see that if u ∈ E (in weak sense) then u is a nontrivial critical point of J(u).Hence, we get (2.12) As a consequence of Lemma 2.6, E d is a nonempty set.We now define stable set W and unstable set U as in [15,27].
By continuity of the functionals I and J on W Some properties of W and U are listed below.
Proof.(i) Let u ∈ W with u = 0, then it follows from the definition of W and (2.5) that for γ > p.In the case γ = p, we also deduce from (2.5) that On the other hand, by virtue of logarithmic Sobolev inequality, we get By choosing µ < p m −1 , it follows from (2.15)-(2.17) that ∇u p p < C d , where C d independent of u.The remain part of (i) can be prove similar to Lions [20].Hence, we possess (i).
(ii) By contradiction, we assume that 0 ∈ U .Then there exists a sequence {u n } ∈ U such that u n → 0 in W 1,p 0 (Ω) as n → ∞.It follows from (i) that u n ∈ W for n sufficiently large.This contradicts to the fact that W ∩ U = ∅.
(iii) It is clear that E d ⊂ N .We now let u ∈ W ∩ U , then I(u) = 0 and J(u) ≤ d.Since (ii), we get u = 0 and therefore u ∈ N .On the other hand, by variational characterization of d, one has This implies u ∈ W ∩ U .The lemma has been proven.

Main results
Firsly, we introduce the definitions of weak solutions to (1.7) and maximal existence time.
satisfies the initial value u(0) = u 0 and the equation (1.7) in a generalized sense, that is, for all v ∈ L p 0, T; W 1,p 0 (Ω) .

Definition 3.2 (Maximal existence time).
Let u be a weak solution to problem (1.7).Then we define the maximal existence time T max of u as follows: • if u := u(t) exists on [0, T) for all T > 0, then T max = +∞.In this case, we say that u is a global solution of (1.7); • if there is T > 0 such that u := u(t) exists on [0, T), but it does not exist at t = T, then T max = T.In this case, we say that u is blow up at t = T.
We now give the existence and nonexistence of global weak solutions to (1.7) depending on parameters m, p and q.Theorem 3.3.Let T > 0, u 0 ∈ W 1,p 0 (Ω) and let m, p be constants satisfying (1.4).Then we possess the following statements.
Next, we give similar results as in [15,27,31] on the existence and nonexistence of global solution when the initial data u 0 is in stable set W and unstable set U .Theorem 3.4 (Global existence for J(u 0 ) < d).Let m, p satisfy (1.4) and q > 2 such that p ≤ (m − 1)(q − 1) and u 0 ∈ W. Then the problem (1.7) admits a global weak solution u ∈ L ∞ 0, T; W 1,p 0 (Ω) with ∂ t U ∈ L 2 (Q T ) and u(t) ∈ W for t ∈ [0, T) for any T > 0. In addition, we have the following decay estimates: for some ω > 0.
Theorem 3.5 (Blow up for J(u 0 ) < d).Let m, p satisfy (1.4) and q > 2 such that p ≤ (m − 1)(q − 1) and u 0 ∈ U .Then weak solution u of the problem (1.7) blows up in finite time, that is, there is T * such that lim t→T * ∇u(t) p p = +∞.
Finally, we have a threshold result on the existence and non-existence of global weak solution to (1.7) in the case J(u 0 ) = d.Theorem 3.8.Let m, p satisfy (1.4) and q > 2 such that p ≤ (m − 1)(q − 1) and u 0 ∈ W 1,p 0 (Ω) \{0} with J(u 0 ) = d.Then the local weak solution u of (1.7) is global provided that I(u 0 ) > 0 and blows up in finite time provided that I(u 0 ) < 0.Moreover, in the former case, there exists a positive constant ω 1 such that for some t 1 > 0.

Proof of Theorem 3.3
In this section we prove the existence of weak solutions by Faedo-Galerkin method.The proof comprises of several steps in which we use the following well-known Gronwall-Bellman-Bihari integral inequality [1, p. 53].
Lemma 4.1 (Gronwall-Bellman-Bihari). Let S(t) be a nonnegative continuous function such that where C 1 , C 2 are positive constants.Then we get (ii) S(t) ≤ C 1 exp{C 2 t} for κ = 1; Step 1: Finite-dimensional approximations Let w j be a system of basis functions in W 1,p 0 (Ω) and define Let u 0k be an element of V k such that as k → ∞.We shall construct the approximate solutions u k (x, t) of the problem (1.1) by the form where the coefficients α kj (1 ≤ j ≤ k) satisfies the system of integro-differential equations for i = 1, 2, . . ., k, with the initial conditions In order to recognize that the system (4.3)-(4.4) has a local solution, for α = (α 1 , α 2 , . . . , Then it is obvious that the system (4.3)-(4.4)can be rewritten as which is also equivalent to the integral equation where α k (t) = (α k1 (t) , α k2 (t) , . . . ,α kk (t)) T .The standard theory of ordinary differential and integral equations yields that there exists a positive 0 Step

2: The fundamental priori estimates
In order to obtain the boundedness of the approximate solutions {u k }, we need the following inequality.
Lemma 4.2.Let 1 < m < p < ∞ and r be a constant such that p ≤ r < p 1 + m n if p < n and p ≤ r if p ≥ n.
Then for each ε > 0, there exists a positive constant C ε such that ) Proof.By virtue of Gagliardo-Nirenberg inequality, we have , with α = θr p .
Multiplying both sides of (4.3) by α ki (t) and taking the sum over i = 1, 2, . . ., k, and then integrating with respect to time variable from 0 to t, one has where We now estimate I (u k (t)).By elementary inequality, we get the following estimate for β > 0 sufficiently small We now consider the two following cases: Case 1: (m − 1)(q − 1) < p − 1.In this case, we have γ < p.By virtue of Young inequality and Poincaré inequality, we get with ε > 0. It follows from (4.9) and (4.11) that By choosing ε = p−1 p(m −1) , we deduce from (4.1), (4.8) and above inequality that If this is the case, then we have p ≤ γ < p 1 + m n .By Lemma 4.2 and (4.10), we derive that where κ > 1 and ε > 0, which implies By choosing ε = p−1 p(m −1) , it follows from (4.1), (4.8) and (4.15) that where κ > 1 and C 1 , C 2 are positive constants independence of k, and By virtue of Gronwall-Bellman-Bihari integral inequality, Lemma 4.1, there exists a constant Now, by multiplying the i th equation of (4.3) by α ki (t), summing up with respect to i and integrating with respect to time variable from 0 to t, we obtain where Thanks to (4.1) and the continuity of J, there is a positive constant C such that We now estimate J (u k (t)).It is worth noting that On the other hand, it follows from (4.12)-(4.13)and (4.15)-(4.18) that for sufficiently small ε > 0. Hence, we get for sufficiently small ε > 0.
Step 3: Passage to the limit In this section, we use some compactness results which is given by Matas and Merker [23].
From the priori estimates devired above (see (4.13), (4.18) and (4.22)), we deduce a subsequence that still denotes as {u k } such that u k → u weakly in L p 0, T; W 1,p 0 (Ω) , (4.23) ∆ p u k → ∆ p u ex weakly in L p 0, T; W −1,p (Ω) .4.24) that ϕ (u k ) is bounded in L ∞ (0, T; L m (Ω)) and is relatively compact in L 1 (0, T; L m (Ω)).By monotone operator theory, using similar arguments as in [23], we get (ϕ(u)) ex = ϕ(u) and This implies On the other hand, direct computation gives us By the Poincaré inequality and Lemma 4.2, it is not difficult to see that Combining (4.29) and (4.30), we get 3), we obtain for all w ∈ L p 0, T; W Here we use the well-known Hölder and Young inequalities.If 1 < m < 2, then we have Step 4: Energy estimate Similar to the method in [20,26,31], let Θ be the function which lies in C[0, T] and is nonnegative.We deduce from (4.19) that Let k → ∞, the right-hand side of this equality tends to T 0 J (u 0 ) Θ(t)dt and using the lower semi-continuous with respect to the weak topology of L p 0, T; W Since Θ is arbitrary, this inequality implies (3.2).

Proof of Theorem 3.4
Step 1: Global existence As in the proof of Theorem 3.3, since u 0 ∈ W, we can find a sequence of Faedo-Galerkin approximation solutions {u k } such that and satisfies the following identities From (5.1) and the continuity of J, it follows from (5.2) that Next, we shall show that u k (t) ∈ W for all t ∈ [0, T max ) for k sufficiently large.Indeed, by contradiction, we assume that there exists t 0 ∈ (0, On the other hand, thanks to (5.3), we must have I (u k (t 0 )) = 0 which implies u k (t 0 ) ∈ N and therefore This contradicts to (5.3).Hence, we get u k (t) ∈ W for all t ∈ [0, T max ).From this fact and (5.2), we arrive at On the other hand, by virtue of Lemma 2.7, one has (5.4) In addition, since I (u k (t)) ≥ 0, we deduce from (2.5) that Hence, (5.3) leads to This inequality allows us to take T max = T for arbitrary T > 0. The rest of the proof is similar to the proof of Theorem 3.3.Hence, u is a global solution of (1.7) and u(t) ∈ W for t ≥ 0.
Step 2: Decay estimates We shall need the following lemma.
We first construct subsets of W which are invariant under the flow of (1.7).For any ε ∈ (0, d), let Since the boundedness of W, we get immediately that for any ε ∈ (0, d), the set W ε is closed and bounded.In addition, the invariant of W ε under the flow of (1.7) is given by the following lemma which its proof is just a consequence of Step 1. Lemma 5.2.Suppose parameters m, p and q satisfy conditions in Theorem 3.4.Furthermore, assume that ε ∈ (0, d) and u 0 ∈ W ε .Then the local solution u(t) of (1.7) is global and u(t) ∈ W ε for t ≥ 0. Since u 0 ∈ W ⊂ W ε for ε > 0, it follows from Lemma 5.2 that u(t) ∈ W ε for t ≥ 0.
It is worth noting that if there exists first T > 0 such that I (u(T)) = 0, then we get a contradiction Hence, one must have I (u(t)) > 0 for t > 0. On the other hand, by Lemma 2.5, there exists λ * > 1 such that I (λ * u(t)) = 0 which implies (5.5) While I (u(t)) > 0 also implies Combining (5.5) and (5.6), we obtain λ γ * ≥ d/J(u 0 ) > 1.By this fact and the following identity we get On the other hand, by virtue of parametric form of logarithmic Gagliardo-Nirenberg inequality, we obtain with noting that γ ≥ p since (m − 1)(q − 1) ≥ p − 1.By choosing µ = γp/2r(m − 1) and r = p/2, we derive from (5.6) and (5.8) that (5.9) It follows from (5.7) and (5.9) that there is a positive constant C such that The proof is complete.

Proof of Theorem 3.5
First, we need the following lemma which its proof is similar to [15,26,27,31].So we omit it.Lemma 6.1.Let m, p and q satisfy conditions in Theorem 3.5 and u 0 ∈ U , then weak solution u(t) to problem (1.7) satisfies u(t) ∈ U , for t ∈ [0, T max ) .
We next give the proof of Theorem 3.5.By contradiction arguments, we assume that u(t) is global solution, that is, T max = +∞.Then we define the function F : [0, +∞) → R + by A direct computation yields Since γ ≥ p and u 0 ∈ U , by Lemma 6.1, we get u(t) ∈ U for all t ≥ 0 which implies u(t) = 0 and I (u(t)) < 0 due to Lemma 2.7.On the other hand, by virtue of Lemma 2.5, there is λ * ∈ (0, 1) such that I (λ * u(t)) = 0.As a consequence, one has Combining (2.5), (3.2), (6.2) and (6.3), we obtain Since J (u 0 ) ≤ d, this implies that F (t) is an increasing function.In addition, we have We deduce from (6.1), (6.4) and (6.5) that In addition, it follows from (6.5) that for all t ≥ 0. (6.7) We now fix t 0 > 0 and define the function Here T is chosen sufficiently large.Then we can derive form (6.6)-(6.7)that Since m < p ≤ γ, we have γ/m > 1.By setting y(t) = G(t) − γ−m m , this inequality implies y to be a concave function on [t 0 , T].Hence, y(t) reaches zero in finite time, that is, there is T * > 0 such that lim The proof is complete.
Hence, the proof is complete.
We now prove the Theorem 3.8 by two following steps.
As in the proof of Theorem 3.4, we obtain a global weak solution u such that u(t) ∈ W for t ≥ 0.
Hence, u(t 1 ) ∈ W. If we take t = t 1 as the initial time, then by analogous arguments in Step 2 in the proof of Theorem 3.4, we possess (3.7).
This implies u(t) m > 0 and U t (t) 2 > 0 for 0 ≤ t < T max .As a results, t 0 U τ (τ) 2 2 dτ is strictly positive for all 0 ≤ t < T max .Taking t 2 ∈ (0, T max ), then I (u(t 2 )) > 0 and Hence, u(t 2 ) ∈ U .If we take t = t 2 as the initial time, then by using similar arguments as in the proof of Theorem 3.5, we imply that weak solution u(t) of the probelm (1.7) blows up in finite time.

Definition 3 . 1 .
A function u is said to be a weak solution of problem(1.7