Some remarks on quasilinear parabolic problems with singular potential and a reaction term

Some remarks on quasilinear parabolic problems with singular potential and a reaction term

Nonlinear Differ Equ Appl 21 (2014), 453–490 c 2013 Springer Basel  1021-9722/14/040453-38 published online November 10, 2013 DOI 10.1007/s00030-013-0253-y Nonlinear Differential Equations and Applications NoDEA Some remarks on quasilinear parabolic problems with singular potential and a reaction term Boumediene Abdellaoui, Sofiane E.H Miri, Ireneo Peral and Tarik M Touaoula Abstract In this paper we deal with the following quasilinear parabolic problem ⎧ up−1 ⎪ ⎪ + uq + f, u ≥ in Ω × (0, T ), ⎨ (uθ )t − ∆p u = λ |x|p on ∂Ω × (0, T ), ⎪ ⎪ u(x, t) = ⎩ x ∈ Ω, u(x, 0) = u0 (x), where θ is either or (p − 1), N ≥ 3, Ω ⊂ IRN is either a bounded regular domain containing the origin or Ω ≡ IRN , < p < N , q > and u0 ≥ 0, f ≥ with suitable hypotheses The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions In the case of nonexistence result, we analyze blowup phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power Mathematics Subject Classification (2010) 35B05, 35K15, 35B40, 35K55, 35K65 Keywords Quasilinear parabolic problems, Hardy potential, Harnack inequality, Existence, Blow-up, Qualitative properties This work was partially supported by project MTM2010-18128, MINECO, Spain B Abdellaoui, S E H Miri and T M Touaoula are partially supported by a project PNR code 8/u13/1063, Algeria The first author is also partially supported by a grant form the ICTP centre of Italy 454 B Abdellaoui, S E H Miri, I Peral and T M Touaoula NoDEA Introduction This paper deals with the following quasilinear parabolic problems ⎧ up−1 ⎪ ⎪ ⎨ (uθ )t − ∆p u = λ p + uq + f (x, t) |x| u ≥ and u(x, t) =0 ⎪ ⎪ ⎩ u(x, 0) = u0 (x) in ΩT ≡ Ω × (0, T ), on ∂Ω × (0, T ), in Ω, (1.1) where Ω is either a bounded regular domain containing the origin or Ω = IRN , < p < N , q > 0, θ is either θ = or θ = (p − 1), λ ≥ 0, f and u0 are nonnegative under additional hypotheses that we will specify bellow These problems are related to the following classical Hardy inequality ΛN,p  RN where ΛN,p =  |φ|p dx ≤ |x|p N −p p p  |∇φ|p dx, for all φ ∈ W 1,p (RN ), (1.2) RN is optimal and is not attained The same inequality holds in the space W01,p (Ω) where Ω is any domain containing the origin with the same optimal constant As in the case of W 1,p (RN ), the optimal constant ΛN,p is not achieved in W01,p (Ω) We refer to [28] for more details about the Hardy inequality Quasilinear parabolic equations are a classical subject widely studied in the literature They are used in models involving nonlinear diffusion that appear, for instance, in Newtonian Fluid Mechanic, diffusion of high temperatures, some biological phenomena, etc The common fact for these class of operators is that all the rheological laws considered are assumed to verify the so called power law We refer, for instance, to [31,35,38] and the references therein for more details about the use of this class of operators It is worthy to point out that in general these kind of modelizations give only a rough approach to the real problems, but in many times this is sufficient to understand some qualitative behaviors such as finite time blow-up in a suitable sense, finite time extinction, finite speed of propagation, etc Existence and asymptotic behavior of solution to problem (1.1) without the singular term are well studied in the literature, we refer to [6,9,21,29], and mainly the book [37] and the references therein For λ < 0, without any reaction term, existence of global solution was obtained in [34], it is clear that, in this case, the singular term has a regularizing effect on the problem and then a priori estimates are easily obtained In the present paper, we deal with the case λ > 0, under the presence of a reaction term In this case, the situation is more delicate and needs to be carefully analyzed Precedents with the Hardy potential in this framework are, among other references [4,5,19,28] Vol 21 (2014) Quasilinear parabolic problems 455 1.1 Some previous results If p = 2, without the reaction term uq , problem (1.1) has been studied in [10] The authors prove existence and nonexistence results related to the fact that λ ≤ ΛN,2 or λ > ΛN,2 , respectively The semilinear case with λ > 0, under the presence of the reaction term uq , was studied in [3] The authors prove the existence of a critical exponent q+ (λ) depending only on λ such that existence holds if and only if q < q+ (λ) The existence of a Fujita type exponent was also proved Notice that the proofs in the semilinear cases are based on the behavior of the solution near the origin x0 = which is obtained using a suitable change of function In the quasilinear case, this method fails and must be modified The problem (1.1) with p = and θ = was considered by several authors The case λ > and θ = was studied in [28] where the problem is analyzed without the presence of the reaction term In [4], the authors prove that if p < N2N +2 , then there exists a global solution for all λ > under suitable conditions on f and u0 The extension of the above results to quasilinear parabolic problems with Caffarelli–Kohn–Nirenberg weights without reaction term, was obtained in [19] For θ = and λ = 0, concerning the doubly nonlinear problem, we can quote the paper [24] where the one dimensional case is studied without the reaction term For θ = p − 1, a classical Harnack inequality is established in [32] for the homogeneous equation This property will be very relevant for our study 1.2 Main results and organization of the paper We start in Sect by giving some auxiliary results related to the quasilinear associated elliptic problem In particular we prove the existence of a critical exponent q+ (p, λ) which is the threshold for existence, that is, there is no solution, in a suitable sense, for q > q+ (p, λ) and there exists a solution if q < q+ (p, λ) under some convenient conditions on the data In Sect we will consider the case θ = p − 1, which will be called the doubly nonlinear parabolic problem The main result in this part is to study existence versus complete and instantaneous blow-up according to q < q+ (p, λ) or q > q+ (p, λ) respectively We must overcome some difficulties produced by the nonlinearity of the differential operator This is a significative difference with respect to the heat equation In Sect we analyze the Cauchy problem, namely Ω = RN , related to the doubly nonlinear parabolic equation, and q < q+ (p, λ) In Sect 4.1 we prove one of the main results in the paper We find an exponent F (p, λ) such that the Cauchy problem has a family of subsolutions blowing up in a finite time with respect to a suitable weighted norm if p − < q < F (p, λ) < q+ (λ) If F (λ) < q < q+ (λ), we find a global in time family of supersolutions These two results show that F (p, λ) is the Fujita exponent for λ ≥ Notice that this result is also new for λ = The Fujita exponent for the case θ = and λ = was obtained in [26] 456 B Abdellaoui, S E H Miri, I Peral and T M Touaoula NoDEA The case θ = and p < is studied in Sect 5, in this case the strong maximum principle does not hold in general We start by proving the existence of solution if q+ (p, λ) ≤ q ≤ 1, under suitable hypotheses on u0 and f This make the difference between the cases p = and p < One of the main results in this section is to prove that the critical value of the reaction power to have extinction in finite time is q = p − Under this value there exists a positive global in time solution with null initial value, while if p − < q ≤ 1, we are able to prove extinction in finite time In Sect we give some results for the case p > and θ = Here also one of the more relevant facts is to prove that the threshold of the reaction power to have finite speed of propagation, is q = If q < 1, we find a positive solution with zero initial value and we have finite speed of propagation, at least, for < q < p − 1.3 Some preliminaries To deal with quasilinear equation problems we precise the sense in which a solution is considered Namely we have the following definitions Definition 1.1 We say that u ∈ T01,p (Ω×(0, T )) if Tk (u) ∈ Lp ((0, T ); W01,p (Ω)) for all k > where Tk (u), the truncating, is defined by Tk (u) = max{min{k, u}, −k} For simplicity of typing, we set b(s) = |s|p−2 s (1.3) Definition 1.2 Assume that F ∈ Lp (ΩT ) and b(u0 ) ∈ Lp (Ω), where b is defined in (1.3) We say that u is a weak solution to the following doubly nonlinear parabolic problem ⎧ in Ω × (0, T ) , ⎨ (b(u))t − ∆p u = F u(x, t) = on ∂Ω × (0, T ) , (1.4) ⎩ b(u(x, 0)) = b(u0 (x)) in Ω, ′ ′ if b(u) ∈ C([0, T ), Lp (Ω)), (b(u))t ∈ Lp ([0, T )); W −1,p (Ω)) and for every φ ∈ C01 (Ω × [0, T ]) with φ(x, T ) = 0, we have     − F φ b(u0 (x))φ(x, 0)dx+ |∇u|p−2 ∇u, ∇φ = b(u)φt + ΩT ΩT Ω ΩT (1.5) If (F, b(u0 )) ∈ L1 (ΩT ) × L1 (Ω), we can define the entropy solution to problem (1.4) Definition 1.3 Assume that (F, b(u0 )) ∈ L1 (ΩT ) × L1 (Ω) We say that u is an entropy solution to (1.4) if b(u) ∈ C([0, T ]; L1 (Ω)), Tk (u) ∈ Lp ((0, T ); W01,p (Ω)) for all k > and for every φ ∈ Lp ((0, T ); W01,p (Ω)) ∩ ′ ′ L∞ (ΩT ) with φt ∈ Lp ((0, T ); W −1,p (Ω)) + L1 (ΩT ), it holds that Vol 21 (2014)  Quasilinear parabolic problems Θk (b(u(T ))−φ(T ))+ Ω =−   ΩT T φt , Tk (b(u) − φ) + 457 |∇u|p−2 ∇u · ∇Tk (b(u) − φ)  ΩT F Tk (b(u)−φ)+  Ω Θk (b(u0 ) − φ(0) Remark 1.4 For problem (1.1), we say that u is an entropy solution to (1.1) if: up−1 i) λ p , uq , f ∈ L1 (ΩT ) and |x| up−1 ii) (1.5) holds with F ≡ λ p + uq + f |x| The following Theorem is proved, for instance, in [8,11,14,18] Theorem 1.5 Let Ω be a bounded regular domain Assume that (F, b(u0 )) ∈ L1 (ΩT ) × L1 (Ω), then problem (1.4) has one and only one entropy solution u such that b(u) ∈ C([0, T ], L1 (Ω)) As a consequence of the monotonicity of b and the operator, we get the following comparison principle Lemma 1.6 Let u, v be entropy solutions of (1.4) with data (f, u0 ) and (g, v0 ) respectively Suppose that g ≤ f and v0 ≤ u0 , then v ≤ u To overcome the difficulty caused by the nonlinearity of the principal part of the operator, we will use the next extension of Picone inequality, see [1,7] Theorem 1.7 (Picone inequality) Let v ∈ W01,p (Ω) be such that v ≥ Suppose that −∆p v is a bounded Radon measure with −∆p v ≥ and −∆p v = Then for all u ∈ W01,p (Ω),  |∇u|p dx ≥ Ω  Ω |u|p v p−1 (−∆p v) dx As a direct application of the Picone inequality, we have the next comparison principle proved in [1] that extends the classical comparison result obtained by Brezis-Kamin [17] Theorem 1.8 (Comparison Principle) Assume that < p and let f be a nonf (x, s) negative continuous function such that p−1 is decreasing for s > Suppose s that u, v ∈ W01,p (Ω) are such that Then u ≥ v in Ω −∆p u ≥ f (x, u), −∆p v ≤ f (x, v), u > in Ω, v > in Ω (1.6) 458 B Abdellaoui, S E H Miri, I Peral and T M Touaoula NoDEA Analysis of the elliptic equation and some auxiliary results Next, we will give the results that we need about the stationary problem, we refer to [36] for more details ⎧ up−1 ⎪ ⎪ ⎨ −∆p u = λ p + uq + g in Ω, |x| (2.1) u>0 in Ω, ⎪ ⎪ ⎩ u=0 on ∂Ω, where Ω is a bounded domain in RN such that ∈ Ω and g is a nonnegative function with suitable hypotheses To analyze the behavior of the solution to problem (2.1), we follow closely the arguments used in [2,16] Let us look for radial solutions to the following associated elliptic problem −∆p w = λ by setting w(x) = C |x| functions −α wp−1 p , |x| (2.2) Using the expression of the p-laplacian for radial −1 p−2 ′ N −1 ′ (|w′ | wr ) rN −1 we get the next algebraic equation −∆p w = h(α) ≡ (p − 1) αp − (N − p) αp−1 + λ = (2.3) If λ < ΛN,p , the Eq (2.3) possesses exactly two solutions α1 < N −p < α2 p See the details in [2] If λ = 0, then α1 = and α2 = N −p p Indeed, N −p with h( N p−p ) = p α1 = α2 = α= (2.4) N −p p−1 , while if λ = ΛN,p , in this case, the unique strict minimum of h is We start by proving the following local behavior result 1,p (Ω) is a nonnegative supersolution to probLemma 2.1 Assume that u ∈ Wloc lem (2.2), then if u ≡ 0, there exists a positive constant C and a small ball Br (0) ⊂⊂ Ω such that u ≥ C|x|−α1 in Br (0) (2.5) where α1 is defined above Proof The proof is a simple modification of the arguments used in [2], for the reader convenience we include here a short proof Notice that, since u  0, then by the strong maximum principle we obtain that u ≥ η > in a small ball Br (0) ⊂⊂ Ω Fix Br (0) and consider the problem ⎧ wp−1 ⎨ in Br (0) , −∆p wn = λ n−1 p (2.6) |x| ⎩ on ∂Br (0) wn = η Vol 21 (2014) Quasilinear parabolic problems 459 with w0 ≡ By the iteration process and comparison principle, we obtain u ≥ wn On the other hand, it is not difficult to show that wn converges to w, the solution of the problem ⎧ wp−1 ⎨ −∆p w = λ in Br (0) , p (2.7) |x| ⎩ w=η on ∂Br (0) Notice that we can choose C > such that w1 (x) = C|x|−α1 , is also a solution to (2.7) It is clear that w ≤ w1 , to show that w = w1 , we follow closely the argument used in [1] Then we get −∆p w1 −∆p w p−1 − w p−1 = w1 Using (w1 − w)+ as a test function in the previous equation and following the arguments in [1] we reach that (w1 − w)+ = Thus w = w1 and then −α1 u ≥ C |x| in Br (0) (2.8) In particular any entropy positive solution is not bounded  Assume < λ < ΛN,p , we look now for radial solutions to the elliptic equation −∆p w = λ wp−1 q p +w , |x| x ∈ IRN (2.9) Then by setting w(x) = A|x|−α , it follows that Ap−1 ((p − 1) αp − (N − p) αp−1 − λ)r−α(p−1)−p = Aq r−αq , (2.10) p , q > p − 1,, A = ((p − 1)αp − so by identification, we have α = q − (p − 1) N −p wp−1 Since (N − p)αp−1 − λ) q−(p−1) and α < ∈ L1loc (Ω) and wq ∈ p−1 |x|p L1loc (Ω), then by the result of Lemma 2.1 we obtain that α1 < α < α2 which is equivalent to p p +p−1 and q± (p, λ) are well defined If λ = we recover q− (p, 0) = (p−1)N N −p and q+ (p, 0) = ∞ Since we are considering an equation with right hand side in L1 , then we will use the concept of entropy solutions as in Definition 1.3, we refer to [12] for more details Let prove now the following nonexistence result Theorem 2.2 Assume that q > q+ (p, λ) ≡ ((p − 1) + (2.1) has no positive entropy solution p α1 ) Then the problem 460 B Abdellaoui, S E H Miri, I Peral and T M Touaoula NoDEA Proof We argue by contradiction Let u be an entropy solution to (2.1), then using an approximation argument as in [1] we get the existence of a minimal entropy solution obtained as a limit of approximating problems We denote u the minimal solution Let ϕ ∈ C0∞ (Br (0)), then applying the Picone inequality as in Theorem 1.7, for u and ϕ, it follows that     p −∆p u p |ϕ| p p |∇ϕ| ≥ |ϕ| ≥ λ + uq−p+1 |ϕ| p p−1 u |x| Br (0) Br (0) Br (0) Br (0)   p p |ϕ| |ϕ| ≥λ p + α1 (q−p+1) Br (0) |x| Br (0) |x|  p |ϕ| ≥ α1 (q−p+1) Br (0) |x| If q > q+ (p, λ), then α1 (q − p + 1) > p, thus we get a contradiction with the  Hardy inequality (1.2), hence the non existence result holds Remark 2.3 If p = a stronger non existence result is proved in [16] For p = the existence results and the behavior of the positive solution near the singular point can be seen in [36] Existence and nonexistence result for the homogeneous doubly nonlinear parabolic problem with a reaction term In this section we deal with the following doubly nonlinear parabolic problem ⎧ p−1 ⎪ ⎪ up−1 − ∆p u = λ u p + uq in Ω × (0, T ) , ⎪ ⎪ t ⎨ |x| (3.1) u(x, t) > in Ω × (0, T ) , ⎪ ⎪ u(x, t) = on ∂Ω × (0, T ) , ⎪ ⎪ ⎩ in Ω, u (x, 0) = u0 (x) with λ < ΛN,p In [32] it is proved that the operator (up−1 )t − ∆p u satisfies the following classical Harnack inequality Lemma 3.1 (The classical Harnack Inequality) Let u be a positive weak supersolution to the problem (up−1 )t − ∆p u = in R = Bδ (x0 ) × (t0 − β, t0 + β) ⊂ Ω × (0, T ) 1,p in the sense that u ∈ Lp (0, T, Wloc (Ω)) and for all ϕ ∈ C0∞ (ΩT ), ϕ ≥ 0, we have   up−1 ϕt + |∇u|p−2 ∇u∇ϕ ≥ − ΩT ΩT Then, there exists a positive constant C = C(N, δ, t0 , β) such that Vol 21 (2014) Quasilinear parabolic problems  u ≤ C ess inf u, R+ R− − with R = B δ (x0 ) × (t0 − 461 β, t0 − β) and R+ = B δ (x0 ) × (t0 + 41 β, t0 + β) Notice that, if λ = 0, by direct calculation, a selfsimilar solution to the equation, (up−1 )t − ∆p u = in IRN , (3.2) is easily obtained, more precisely, for p > 1, we find that the selfsimilar solution is given by,   p p−1 N p−1 |x| − p(p−1) v(x, t) = t (3.3) e − p t p p−1 In the case where λ > 0, we have the following results Lemma 3.2 Assume that u0  and let u be an entropy positive solution to problem (3.1) in the sense of Definition 1.3, then u(x, t) → ∞ as |x| → for all t > Proof Using the weak Harnack inequality obtained in [32], it follows that u ≥ C in Br (0) × (t1 , t2 ) for all t1 > Notice that C1 r = −∆p log p, |x| |x|   a r , it follows that then by setting v (t, x) = ε (t − t1 ) log |x| ⎧ (p − 1)cp |D(cs)|p−2 K(cs) + cp−1 N s−1 |D(cs)|p−2 D(cs) + p−1 p csD(cs) λ +(θ(p − 1) + p ) + Aq−(p−1) φq−(p−1) (cs) ≤ s for all s ≥ Hence we get the existence of a family of globally defined in time supersolutions Notice that   p   p p−1 c p−1 − q−(p−1) −γ |cx| e − |x| p−1 w(x, 0) = AT p p p−1 T p−1 Consider the set A= u0 ∈ L1 (IRN ) : such that u0 (x) ≤ |x|−γ   p p−1 p−1 for some D > |x| ×e −D p p p−1 It is clear that for all u0 ∈ A, we can find T > and c < close to such that w(x, 0) ≥ u0 (x) Thus w is a supersolution to problem (4.2) Since v(x, t) = is a strict subsolution, then by an iterative argument we get the existence of  a global positive solution u to problem (4.2) Remark 4.6 Assume that p − < q < q+ (p, λ) and consider an initial datum such that u0 (x) ≥ φ, where φ ∈ C0∞ (IRN ) is a non negative function with supp(φ) ⊂ B0 (R) and   φp 1 |∇φ|p − λ p dx (4.15) φq+1 dx > q + IRN p IRN |x| If u is a positive solution to problem (4.2) with datum u0 (x), then there exists T ∗ < ∞ such that  up (x, t)dx → ∞ as t → T ∗ (4.16) BR (0) To proof (4.16), we proceed by contradiction We consider the solution v with datum φ, then by comparison principe we conclude that u ≥ v Hence if (4.16) does not hold we reach a contradiction with (4.15) Existence result for the case θ = and p < 2: solution with and without finite time of extinction In this section we consider the problem ⎧ up−1 ⎪ ⎪ ⎨ ut − ∆p u = λ p + uq |x| u(x, t) = ⎪ ⎪ ⎩ u (x, 0) = u0 (x) , in Ω × (0, T ) , on ∂Ω × (0, T ) , in Ω (5.1) 472 B Abdellaoui, S E H Miri, I Peral and T M Touaoula NoDEA with p < 2, the main goal of this section is to show that under suitable conditions on u0 , we get existence of a nonnegative solution for some q > q+ (p, λ) This will make the difference between problem (5.1) and problem (1.1) and clarify the role of the classical Harnack inequality in the nonexistence results More precisely, for θ = p − the strong maximum principle for the parabolic equation holds, while for θ = 1, only holds for p = We start by proving the following global existence result Theorem 5.1 Assume < p < N2N +2 Let u0 ∈ L (Ω) be a nonnegative function c such that u0 (x) ≤ p , then for all λ > and q < 1, problem (5.1) has a |x| 2−p nonnegative minimal energy solution u such that u ∈ Lp (0, T ; W01,p (Ω)) p p−1 p ) (N − 2−p ) Proof Without loss of generality we can assume that λ > ( 2−p To obtain the existence result we will show that the above problem has a good supersolution Then the existence result follows by using monotonicity Using a rescaling argument we can assume that Ω ⊂⊂ B0 (1) Define t + T0 2−p Sλ0 (x, t) = A(λ0 ) p |x| where λ0 > λ will be chosen later, then if < p < we know that Sλ0 satisfies (Sλ0 )t − ∆p Sλ0 = λ0 where A(λ0 ) = then  p 2−p p−1 2N N +1 , using the result of [4], Sλp−1 p |x| 2−p (p − N (2 − p)) + λ(2 − p) Fixed λ1 = λ0 +λ, (Sλ0 )t − ∆p Sλ0 = λ Sλp−1 Sλp−1 0 + λ p p |x| |x| Since q < 1, under suitable condition on λ1 and T0 we reach that λ1 Sλp−1 q p ≥ Sλ0 |x| Thus to show that Sλ0 is a supersolution to the problem (5.1), we have just  2−p  T0 to compare the initial conditions It is clear that Sλ0 (x, 0) = λ0 |x| p Choosing λ0 such that λ0 T02−p ≥ c, it follows that Sλ0 (x, 0) ≥ u0 (x) Since Sλp−1 Sλp−1 0 p < N2N + λ , then λ p p ∈ L (Ω × (0, T )) Hence we conclude that +2 |x| |x| Sλ0 is a supersolution to (5.1)