Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: \begin{document}$ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0 where \begin{document}$ p>m+2 $\end{document} , \begin{document}$ m\geq0 $\end{document} . Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution \begin{document}$ u $\end{document} : \begin{document}$ u $\end{document} is global and converges to the unique stationary solution in \begin{document}$ C^1 $\end{document} norm for subcritical boundary value, and \begin{document}$ u_x $\end{document} blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments.

1. Introduction. In this paper, we consider the following problem    u t = u xx + x m |u x | p , t > 0, 0 < x < 1, u(t, 0) = 0, u(t, 1) = M, t > 0, where p > m + 2, m ≥ 0 and M ≥ 0. The initial data u 0 satisfies: and where U (x) is the singular steady state of (1) for critical boundary value M = M c . Problem (1) admits a unique maximal classical solution u = u(t, x), whose existence time will be denoted by T * = T * (u 0 ) ∈ (0, ∞]. Note that we make no restriction on the sign of u or u x . The equation (1) arises from several interesting mathematical and physical fields. The Kardar-Parisi-Zhang(KPZ) equation was first proposed in [24], and it can be considered as the fundamental nonlinear Langevin equation. Here h(t, x) describes the height of the surface at time t above the point x in the reference plane. Let ν∆h be diffusional relaxation with diffusion coefficient ν related to the surface tension. The growth term λ|∇h| 2 corresponds to the deposition of new particles on the surface, and λ accounts for the growth speed and modifies the universal scaling properties of the model if λ = 0. The noise term η(t, x), which relates with molecular collision, is irrelevant.
To account for further surface growth effects, Krug-Spohn [26] considered KPZ equation with more general nonlinearity: where β ≥ 1. The large-scale properties of the surface are governed by scaling exponents β in the stochastic case. Some properties of solutions to this equation have been studied numerically in [2,32]. In this paper, we consider the case λ = x m , which means that, there is a power-like degeneracy at x = 0, and we clarify clearly the effects of the degeneracy on the long time behavior of global solutions.
Considering the case F = F (u, ∇u) with general boundary conditions, Soulpet [37] showed that gradient blowup occurs for suitable large initial data, which solutions are global and bounded in C 1 norm if the initial data are small enough. Many mathematicians pay special attention to (4) in the case F = F (∇u) = |∇u| p , a vast literatures exists [28,40,17,39,3]. It was proved in [40] that gradient blowup can only occur on the boundary and an upper estimate of blowup profile of |∇u| is given. Li and Souplet [28] provided the precise location of gradient blow-up points within the boundary under suitable assumptions on Ω ⊂ R 2 and the initial data u 0 . Guo and Hu [17] obtained that (T − t) − 1 p−2 is the exact blowup rate in the one-dimensional case for time-increasing solutions. The self-similar rate is not the only rate, there are other rates which is usually refereed to as Type II rate, for which we refer the readers to the survey works [10,15,18,19,20,21].
We take into account the problem (1), corresponding to the case F = F (x, u x ) = Alikakos et al. [1] proved that u x blows up in a finite time. When m > 0 for to (1), Zhang and Hu [42] proved that gradient blowup occurs exactly at the single boundary point x = 0 in finite time (see [42,Lemma 3.1]). Meanwhile, the authors derived the blowup rate lower and upper bound estimates: [42,Theorems 4.1 and 5.2]. In [42], conditions are added to u 0 such that u t ≥ 0 and u x ≥ 0, which is not required in the present paper.
As we all know, steady-state solutions are the possible limits as t → ∞ of the corresponding time-dependent solutions if they exist globally. As for the steady states, there are three cases according to the value of M .
, an explicit formula will be given in Section 2. M c is the critical value Further, the steady state V M increases monotonically with respect to M .
Next, a fundamental result is that the solution of (1) satisfies the maximum principle: min min Since the problem (1) is well-posed in C 1 , so only three possibilities can occur: (I) u is global in time and is bounded in C 1 norm: (II) u blows up in finite time in C 1 norm: For results on case (I), we refer to [3,4,40,44,29]. For (1), assuming 0 ≤ M < M c and m = 0, Arrieta et al. [3] derived that u is global and converges in C 1 norm to the unique stationary solution. Recently, Attouchi [4] modified the method used in [3] and extended results to the one-dimensional degenerate Hamilton-Jacobi equation: where q > p > 2. It was showed that u is global and converges in W 1,∞ (0, 1) to the stationary solution. The paper [4] faces a number of additional technical difficulties, caused by the lack of regularity. For higher dimensional case, Li [29] focused on the following problem: where B r,R = {x ∈ R n ; r < |x| < R}, ∂B r = {x ∈ R n ; |x| = r} and p > 2. Li presented that the global solution converges in C 1 (B r,R ) to the unique regular steady state, and obtained uniform exponential convergence rate for subcritical boundary value. Motivated by the results of the works [3,4,29], the present paper considers (1) with the weighted nonlinear gradient term, we get similar results (see Theorem 1.1). Due to the accurate estimates (cf. Section 2), classical Lyapunov argument and regularity results (see Propositions 1 and 2), we proceed by contradiction by showing that any C 1 unbounded global solution would converge to a singular steady state, which does not exist. For the case (III), Souplet and Zhang [40] studied inhomogeneous Hamilton-Jacobi equation: and h is radially symmetric. They showed that there exists λ * ∈ (0, ∞) such that u has gradient blowup in infinite time for λ = λ * . Further, some mathematicians focus on descriptions of infinite growup rate of gradient, and it can be obtained by matching of inner and outer asymptotic expansions. Souplet and Vázquez [39] devoted to studying a heat equation with a nonlinearity depending on the first-order spatial derivatives of u: where p > 2, M ≥ 0. The rate of divergence of u x (t, 0) and the asymptotic behavior of u(t, x) as t → ∞ are precisely obtained for critical boundary value. The paper seems to be the first example of gradient blowup in the infinite time for a semilinear parabolic equation. For higher dimensional case, Li [29] extended results to (6) and obtained the growup estimate: where ν is any normal vector field, and λ 1 is the first eigenvalue of an associated linearized problem. Moreover, the present paper considers the equation (1) with weighted nonlinearity, i.e. x m |u x | p , where p > m + 2 and m ≥ 0. The weighted term x m brings many obstacles, which will be explained later. The matched asymptotic method can also be used to describe blowup profiles of solutions. We refer the readers to the survey papers [16,6,10,31,12,7,13,11,23,25]. Let us now mention some results already known for the semilinear equation Form Galaktionov and King [12], we know that if p = N +2 N −2 and u 0 is positive and symmetric, then as t → ∞, where γ 0 = γ 0 (N ) > 0 is a constant independent of initial data. Related results for the case p ≥ p u = N −2 N −1 , N > 10 were obtained in Galaktionov et al. [7]. It was proved that if u 0 belows the singular steady state U s (x), then as t → ∞, Meanwhile, [7] also focused on the semilinear Frank-Kamenetskii equation It showed that if N > 10 and u 0 belows the singular steady state U s (x), then where α 0 = α 0 (N ) > 0. Next, Galaktionov and King [13] considered the phenomenon of critical dimension N = 10, they derived that where α 0 is given by the first eigenvalue of the associated linear differential operator.
The method of matched asymptotic expansions can also be applied to other PDE models, see previous studies [23,25,35]. Ju et al. [23] studied the quasi-neutral limit for the two-fluid isentropic Euler-Poisson system in multi-dimension. For the well-prepared initial data, they formally derived the compressible Euler equations, and this formal limit is very different from the unipolar case for which the limit is the incompressible type Euler equations. Kavallaris and Souplet [25] considered a special case of the Patlak-Keller-Segel system in a disc, which arises in the modeling of chemotaxis phenomenon. For a critical value of the total mass, the solutions are global but density is unbounded, leading to a phenomenon of mass-concentration in infinite time. It is worth noting that the maximum of density behaves like e √ 2t , unlike the conclusions mentioned above, and the growup rate is neither polynomial nor exponential.
For (1), the case M > M c , which corresponds to (II), was studied by Zhang and Hu in [42]. The cases 0 ≤ M < M c and M = M c , which correspond to (I) and (III) respectively, will be considered in the present paper. Clearly, the three possibilities of the solution for (1) are all taken into account.
Our main results are stated as follows.  (2) and (3) hold. Then the solution u of (1) exists globally and satisfies where λ = λ(p, m) > 0 is defined in Proposition 6 as the first eigenvalue of an associated linearized problem, and µ = (m+1)λ p−m−2 . For (1), the works [3] and [39], which had discussed the case m = 0 throughly, are significant references for us to solve problems in the case m > 0. The presence of x m leads some difficulties as follows.
First, for the case m > 0, the stationary solution presents a complicated formula in the integral form (the explicit formula appears at the beginning of Section 2), which poses two major difficulties. One is the convergence proof between steady states satisfying different boundary conditions (see U µ → U as µ → ∞ in Section 4.2). We apply monotone convergence theorem to prove it, avoiding the complicated calculation caused by integral form. The other is in process of matching the inner and outer regions (see the proof of Theorem 4.4). We introduce a sufficiently small parameter to characterize the boundedness of x in the inner region. Finally, a complicated computation can present an inequality with , letting → 0, we can get the asymptotic behavior of u x . Moreover, the remainder term of Taylor expansion is also given by analyzing series convergence. When m = 0, it is easy to calculate because the steady state solution is given in polynomial form.
Second, the presence of x m leads to the failure of the maximum principle. When m > 0, by differentiating (1) with respect to x, the function h = u x satisfies where b(t, x) = px m |h| p−2 h. The maximum principle is obviously false by the presence of the term mx m−1 |h| p . The failure of maximum principle leads to two major difficulties. One is to rule out infinite time gradient blowup at boundary point x = 1 (see Lemmas 3.1 and 3.2). The work [3] (see Lemma 2.6) illustrated the situation with the fact that u x satisfies the maximum principle and attains the maximum on the boundary in the case m = 0. The present paper needs to use more delicate and mathematical techniques than those which suffice for the case m = 0. We construct a Lyapunov functional to obtain the convergence result of steady state W (x) (see Proposition 2). Combining with the fact that W (x) satisfies the maximum principle for we can get the conclusion, avoiding the fact that u x can not attain the maximum on the boundary. The other is that the time-dependent perturbation vanishes as t → ∞ (see Lemma 4.2), the perturbation appears after transformation for (1). We construct appropriate auxiliary function z and use the Hopf lemma to obtain the decaying property of perturbation.
Third, the presence of x m affects the range of k, which is defined as the index of weighed term in an associated linearized problem (51). We perfect the conclusion of eigenvalue problem in case k ∈ [1, 3) (see [3,Proposition 5.1]) to the case k ∈ [1, ∞) (see Remark 4 (ii)), so that we can obtain asymptotic behavior of u in outer region by constructing supersolution and subsolution.
The paper is organized as follows. In Section 2, we give some preliminary estimates and gradient estimate. In Section 3, we consider the case 0 ≤ M < M c and prove Theorem 1.1. In Section 4, we study the case M = M c and prove Theorem 1.2.
2. Preliminary estimates and gradient estimate. Let us recall that the solu- . For given M ≥ 0, (1) admits a unique steady state in the integral form. In special case m = 0, a unique stationary solution of (1) is given in the polynomial form, i.e.
In this section, we will give the following useful lemmas which are similar to the ones (see Lemmas 2.1, 2.2 and 2.4) in [3], the proofs are omitted here.
By using the upper bound of u t , the following lemma provides the upper and lower bounds on u x , in particular, it shows that u x is bounded away from the boundary. and The following lemma provides a lower bound on the blowup profile of u x in case gradient blowup occurs in infinite time near x = 0 and x = 1. and The following lemma will give gradient estimate on the basis of the Bernsteintype arguments and maximum principle, see [40,44,28] for further details.
Lemma 2.4. Let u be the maximal, classical solution of (1), then where Differentiating the equation (16) with respect to x yields that: Multiplying (17) by 2v x , we have Using Young's inequality, we have Next, we start to estimate each term appearing in the inequality (19).
For R → 0, it holds (ii) In [42], Zhang and Hu showed that x = 0 is the only blow-up point of (1) and provided the following gradient estimate: Obviously, the estimate in Lemma 2.4 is more preciser than that in [42].
(iii) The cut-off function η(x) had been constructed in [40]: To explain the proof, we construct a Lyapunov functional, which exists for one-dimensional uniformly parabolic equations. A Lyapunov functional was possessed by Zelenyak in a classical paper [41], and it plays a vital role in the proof of the convergence to steady states.
There exist a pair of functions φ ∈ C 1 (D K ; R) and ψ ∈ C(D K ; (0, ∞)) with the following property.
For any solution u of (1) with |u| ≤ K with the definition Moreover, we have φ ≥ 0.
The above proposition can be proved by slightly modifying the Proposition 3.1 in [3], so we omit the details here. According to Proposition 1, Lemmas 2.1 and 2.2, we would obtain the following convergence conclusion.

This implies that
Introducing the transformation s = t + t n k , it follows from Lemma 2.1 that Since ∂ t u n k → W t as t → ∞ and is arbitrary, we know that W t ≡ 0. Thus, W satisfies (31). Since the sequence {t n k } k∈N is arbitrary and the stationary solution (for given M ) is unique, these imply that the whole solution u(t, x) converges to W (x).
The following lemma is a direct conclusion of the convergence of u to the steady state. Thanks to (12), we can obtain that infinite time gradient may blow up towards −∞ for x = 1, the following lemma can enable us to rule out this case. Proof. Assume that the lemma is false. By (12), there exists a sequence t n → +∞ such that u x (t n , 1) → −∞. Fix > 0, for n ≥ n 0 ( ) large enough, it holds that (−u x ) + (t n , 1) → +∞. Using (14) and p > m + 2 ≥ 2, we deduce for 0 ≤ x ≤ This implies that  Thanks to (11), we can obtain that infinite time gradient may blow up towards +∞ for x = 0. In what follows, we are ready to rule out this case.
Proof of Theorem 1.1. We proceed by contradiction. Assume that u is a global solution of (1) which is unbounded in C 1 . From Proposition 2, we have that as It follows from Lemmas 2.2 and 3.2 that, if u is unbounded in C 1 , then there exists a sequence t n → ∞ such that u x (t n , 0) → ∞.
Using this and (13), we deduce This can imply that W Proposition 4. We have the following estimate: Next in view of Proposition 4, an exponential lower bound of u x (t, 0) has been provided in following proposition. It plays a key role in the inner region analysis.

4.2.
Inner-region analysis. This section provides the asymptotic behavior of u in a small region near x = 0 for large t. This is called inner region (boundary layer) analysis. Following the ideas in [39,7], the solution u(t, x) is given by a quasi-steady problem and it is close to a stationary solution.
Consider the re-normalized stationary equation It has the form U µ (x) = µ γ U 1 (xµ β ) with β > 0, γ = m+2−p p−1 β < 0. Moreover, we can select special range for γ, which is mainly used to obtain gradient growup rate (see the proof of Theorem 4.4). We assume β > 0 and λ = λ(p, m) > 0 is defined in Proposition 6 as the first eigenvalue of an associated linearized problem.
We claim that U µ (x) increases monotonically to U (x) as µ → ∞. In fact, which is deduced from U 1 (x) > 0 for all x > 0. This implies that U µ (x) increases monotonically with respect to µ for µ large enough. In addition, it follows from In other words, U (x) is the solution of (39) corresponding to µ = ∞, which completes the proof of the claim.

CAIHONG CHANG, QIANGCHANG JU AND ZHENGCE ZHANG
For each µ > 0, U µ (x) = µ 2−p U 1 (xµ p−1 ) is the unique solution of Clearly, U µ (x) is given in polynomial form, the result U µ (x) → U (x) as µ → ∞ can be obtained through detailed calculation. In the present paper, U µ (x) has the integral form, which leads to some computational difficulties. So we apply the monotone convergence theorem to illustrate the situation.
The present paper would introduce the rescaling parameter that diverges to ∞ as t → ∞, as we already know from Proposition 5. Furthermore, the fact that α(t) increases monotonically with respect to t can be proved in Theorem 4.1 (i).
, where the constants C, ρ will be determined later. We claim that z x (t, 0) ≤ 0, for t 1.
(45) Differentiating the equation (1) with respect to t: A direct calculation shows that z t = u tt + 2Cρuu t e ρu 2 , z x = u tx + 2Cρuu x e ρu 2 , and z xx = u txx + 2Cρuu xx e ρu 2 + 4Cρ 2 u 2 u 2 x e ρu 2 + 2Cρu 2 x e ρu 2 . It follows from above equalities, (46) and (1) that where C > 0 and ρ < 0 is small enough. The last inequality holds due to F (ρ) < 0. In fact, F (ρ) could be seen as a quadratic function of ρ. The fact, which as t → ∞, indicates that x m |u x | p−2 u ∼ 1 as t → ∞, so the coefficient of ρ is bounded in the term F (ρ). Moreover, the result that the coefficient of ρ 2 is bounded can be provided by (2) and (5). Hence, F (ρ) < 0 is obviously true for ρ < 0 small enough.
From (2), (5) and ρ < 0, we derive that 0 < 1 − e ρu 2 ≤ C , where C > 0. In view of Lemma 2.1, there exists a constant C > 0 that Using (47), (48), z(t, 0) = 0 and the Hopf lemma, we obtain that z x (t, 0) < 0. It follows from (5) that where C > 0. Thus Remark 3. In addition, the auxiliary function z can be given by where C i , C > 0. The term ∞ 0 C i u i can be regarded as Taylor expansion of e ρu 2 . Further, the order of expansion also has corresponding requirements.
In order to prove the main result, we need to introduce the following regular eigenvalue problem, as an approximation to a singular problem (51): for each ∈ (0, 1). We denote λ > 0, ϕ > 0 to be the first eigenvalue and the first eigenfunction of problem (54), respectively. Next, the exponential convergence rate is established by the accurate constant λ just defined in Proposition 6.
Proof. It follows from (56) that where η is small enough and C > 0. Hence By as t → ∞. So, x m |u x | p → +∞ for η small enough. Combining with Lemma 2.1, we get that Clearly, u x is decreasing in x. Letting α = p−m−2 p−1 > 0 and x = x 1 (t) = Ce − λ α t > 0, we conclude that x 1 (t) → 0 as t → ∞. For t large enough, by the monotonicity of u x and (59), we can deduce that u x (t, 0) > u x (t, x 1 (t)) = lim t→∞ u(t, x 1 (t)) − u(t, 0) x 1 (t) In the inner region, since y = xα β is in the bounded sets as t → ∞, there exists ≥ 0 such that x = α −β− , will be determined below. Therefore, (60) is equivalent to Due to γ < 0 and α(t) → ∞ as t → ∞, each term on the right of (61) converges to 0 as t → ∞. Moreover, Then, for (61), every term after the second term converges to 0 faster than the first term. Hence