GLOBAL EXISTENCE AND GRADIENT BLOWUP OF SOLUTIONS FOR A SEMILINEAR PARABOLIC EQUATION WITH EXPONENTIAL SOURCE

Throughout this paper, we consider the equation 
\[u_t - \Delta u = e^{|\nabla u|}\] 
with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.

Then, the nonnegativity of u can be directly obtained by the maximum principle.
For the problem (1), there has some related results in the one-dimensional or radial case. In [17], Zhang and Hu considered the one-dimensional case, and they proved that ∇u will blow up in a finite time, i.e. u(·, t) ∞ < ∞, lim t→T ∇u ∞ = ∞, t ∈ (0, T ) 3020 ZHENGCE ZHANG AND YAN LI for some large u 0 ∈ X. The blowup rate was also established in that paper. In [18], Zhang and Li studied the radial case and obtained some similar results to [17]. In [3], the continuation after blowup and the rate of converging to a singular steady state were obtained in the one-dimensional case. Another important problem is the large time behavior for the global solution. For this problem, Zhu and Zhang gave some results on annular domains in [21]. Besides, the boundedness of the global solution is also an interesting problem. The one dimensional case had been solved by Zhang and Li in [19]. However, there has no result on general domains.
Similar to (1), the problem x ∈ Ω, t > 0, had been studied by many people so far. It's well-known that the differential equation in (2) comes from the famous Kardar-Parisi-Zhang (KPZ) equation which was first mentioned in [6] and developed by Krug and Spohn aiming at studying the influence of the nonlinearity to the solution in [7]. When p > 2, it was shown in [5,10,12,13,14] that the solution of (2) cannot exist globally if the initial data is suitably large. While if there exists > 0, such that the initial data satisfies: u 0 C 1 (Ω) < , then it was shown in [12, Proposition 3.1] that the solution is global and bounded in C 1 (Ω). If the solution of (2) exhibits gradient blowup phenomenon, we want to know where it occurs, what's the exact gradient blowup rate and so on. In [8], Li and Souplet showed that the gradient blowup will occur at a single point of ∂Ω if Ω ⊂ R 2 and has some symmetric properties. For the blowup rate, it had been shown in [4] that the exact rate is ( The lower estimate was also extended to higher dimensional space by the rescaling methods, while the upper estimate in this case is still an open problem. Apart from the finite time gradient blowup, the properties of the global solution is also worth researching. This had been partially solved in [1,14,15] and the references therein. For other similar problems which are related to the gradient blowup, we refer the readers to [2,10,16,20] and the references therein. We are also eager to know whether the results obtained for (2) are still valid for the problem (1). An obvious fact is that (2) is invariant under the self-similar transformation while (1) is not. Hence, we conjecture that the properties for the self-similar solutions of (2) cannot extend to (1). Another important fact is that the differential equations in (1) and (2) are two specific examples of the more general equation u t − ∆u = F (x, t, u, ∇u) for some F ∈ C 1 (Ω × R + × R × R N ) satisfying suitable growth conditions. Some typical results on the general case when F = F (∇u) had been studied by Fila and Lieberman in the one-dimensional case and by Souplet in the high-dimensional case, see [2,12]. In this paper, we are to consider the specific case (1), and to study the large time behavior of the global solution and the finite time gradient blowup.
For the large time behavior, we will consider the relation between the global solution and the stationary solution of the following stationary problem.
For the solution of (3), we will use the following definition.
GLOBAL EXISTENCE AND GRADIENT BLOWUP 3021 Definition 1.1. A solution v of (3) is a function v ∈ C 2 (Ω) ∩ C 0 (Ω) which solves the differential equation in (3) in the classical sense in Ω.
Now, let us introduce our main results for the global solution of (1).
Theorem 1.2. Assume that the solution u of (1) is global in time for some u 0 ∈ X.
Then there exists a solution v of (3) and u must converge uniformly to v in C(Ω), as t → ∞.
Remark 1. As a direct consequence of the maximum principle, one can easily verify that the solution of (3) must be unique if it exists. Souplet and Zhang showed in [15] that the solution of −∆u = |∇u| p with homogeneous Dirichlet boundary condition can only be 0. However, this is incorrect for our problem as any constant can't be the solution of (3). But, if we rewrite the differential equation in (3) as the inhomogeneous one −∆v = e |∇v| − 1, then the only solution must be 0 provided that v = 0 on ∂Ω. Different from the global existence and convergence, the following two theorems give another behavior of the solution of (1).
As was shown in Theorem 1.4, the large initial data can imply the finite time gradient blowup, we are also naturally eager to know whether small initial can imply the global existence and boundedness or not. For example, if there exists some > 0, such that u 0 C 1 < , then the solution of (1) is global and bounded in C 1 (Ω). Our main thought is to construct a supersolution and a subsolution. The subsolution can be obtained by reforming the solution of −∆χ = 1 with χ = 0 on ∂Ω. However, due to the technical difficulty, we are unable to construct a supersolution.
Unlike the condition that u 0 (x) is large enough in certain Ω, the following theorem shows that whether gradient blowup can occur or not depends on the measure of Ω. In [2], Fila and Lieberman construct an example saying: gradient blowup may occur only at x = 0 for any u 0 (x) ∈ C 1 [0, L], where L must be larger than a given L 0 . Here, we develop their result to higher dimensional spaces for some given initial data. More precisely, our result implies that gradient blowup can occur if the measure of Ω is small enough. Theorem 1.5. Let φ 1 be the solution of the following eigenvalue problem where λ 1 is given by λ 1 = inf{ ∇u L 2 (Ω) ; u ∈ K}, K = {u ∈ H 1 0 (Ω); u L 2 (Ω) = 1}. Let u 0 (x) = φ 1 and N ≥ 2, Then there exists a small constant M > 0 such that gradient blowup occurs in finite time if |Ω| < M .

Remark 4.
We also note that the initial data used in Theorem 1.5 can be replaced by some smooth function φ(x) which satisfies: Ω φ(x) dx = 1.

2.
Preliminaries. At the beginning of this section, we bring in the following notations.
• n: the unit exterior normal vector on ∂Ω.
If Ω is unbounded, assume additionally that sup Q T (u 1 − u 2 ) < ∞ and that either Finally assume that, for some A, (ii) Let u be a solution of (1) and v a solution of (3). Then , then the solution of (1) satisfies min 0, ess inf for x ∈ Ω, 0 < t < T * .
Next, we will introduce the following lemma as an application of the comparison principles above. This result will play an important role in the proof of Theorem 1.3.
Then T * (u 0 ) = ∞ and there exists u 0 ∈ X satisfying: T * ( u 0 ) = ∞ and the corresponding solution is increasing in time.

Gradient estimates.
In this section, we will give two different gradient estimates which are based on the Bernstein-type arguments.
is a classical solution of (1) and that u ≤ M in Q T,R . Then in Q T,R/2 for some C > 0.
Let us now estimate each term appearing in the inequality (8).
Since u ≤ M , we have 2 ≤ e v ≤ 2 + M which implies that 2 < e e v |∇v| w − 1 2 , then By the second inequality in (7), we can get where C = 4C. Besides, we have Combining inequality (8) with estimates (9)-(11), and using Young's inequality and the fact that 0 ≤ η ≤ 1, we can get e −e v|∇v| Lz ≤ −2ηw where a ≥ 2 3 , A = C 2 3 1 (1 + R −2 + R −1 ) 2 . Then we can deduce that In order to obtain the estimate (4), it's necessary to construct another auxiliary function φ(t) = ct −2 which satisfies φ (t) ≥ − 1 2 φ 3 2 (t) with suitable c > 0. Similar to [15], we can set z(t) = z(t + t 0 ) − φ(t) for fixed t 0 ∈ (0, T ). Then z(t) As lim t→0 z(t) ≤ 0, we can assert that z ≤ A in Q T −t0,R by Lemma 2.2. Letting t 0 → 0, using the fact that z = η|∇v| 2 and that 0 < δ ≤ η ≤ 1 in Q T,R/2 , we can deduce that Remark 5. The cutoff function η had been constructed explicitly in [15]: If we put v = u instead of v = ln(M + 2 − u), then we have the following gradient estimate which is independent of u and the variable t. Theorem 2.7. Let u be a maximal, classical solution of (1). Then where Proof. For the proof of Theorem 2.7, we just need to modify the last step of the proof of [18, Theorem 3.1] as Remark 6. One can also see from the estimate (12)  We can also derive the following estimate for u from (12).
The solution of (1) satisfies 3. Proof of the results on stationary solution. In this section, we will give the simple proofs of Theorems 1.2 and 1.3 which are based on the parabolic regularity theory.
Proof of Theorem 1.2. By Lemma 2.5, we know that there exists a global solution u which is increasing in time for some u 0 ∈ X and satisfies (1). The estimate (13) implies that there exists a function v ∈ C 0 (Ω) such that lim t→∞ u(x, t) = v(x) for all x ∈ Ω. The next step is to show that v ∈ C 2 (Ω) ∩ C 0 (Ω) and that v is a solution of (3) in the sense of distribution. Hence, v is a classical solution of (3). According to the gradient estimate (12), we assert that |∇ u| ≤ C( ) in Ω × [0, ∞). Thus, e |∇ u| ∈ L ∞ loc (Ω) for t > 0. Combining this with the fact that u ∈ L ∞ loc (Ω) and using parabolic regularity theory, we deduce that there exists 0 < β < 1 such that ∇ u C β,β/2 (Ω ×[τ,τ +1]) ≤ C( ) for each > 0 and τ ≥ 1. Then the interior Schauder estimates imply that where 0 < γ < 1 and > 0.
For each > 0, let u n := u(·, t n + ·) and t n → ∞, by the diagonal procedure and Ascoli-Arzelà theorem, we can find a subsequence, still denoted by u n , such that u n converges in C 2,1 loc (Ω × [0, 1]) to a function v which must coincide with v. Thus, we can deduce that v ∈ C 2 (Ω) ∩ C 0 (Ω).

ZHENGCE ZHANG AND YAN LI
Letting n → ∞, we can get Ω v∆ϕ + e |∇v| ϕ dx = 0, which implies that v is a solution of (3) in Ω in the distributional sense. Therefore, v is a classical solution of (3).
Remark 7. If we assume that the initial data satisfies: ∆u 0 + e |∇u0| ≤ 0, then the solution is decreasing with respect to t. Thus, we can apply Dini's theorem [11,Theorem 7.13] on uniform convergence of monotone sequences to obtain that u converges to v uniformly in C(Ω).
Proof of Theorem 1.3.
If v ∈ C 1 (Ω), this is a direct consequence of Lemma 2.5.
If v ∈ C 1 (Ω), then there holds lim r→R v (r) = −∞ by the estimate (12). In order to obtain the global existence, we may assume that T * < ∞. Our next step is to show Combining this with the fact that we can derive a contradiction which implies the global existence of the solution of (1). For (14), we just need to consider the annular domain B ρ,R := {x ∈ R N ; ρ < r < R} for some ρ > 0. By the same arguments as in [21,Section 4], we can obtain (14).

Remark 8.
It's necessary to point out that the condition for v in [21] is v(ρ) = 0, v(R) = M . However, the condition in Theorem 1.3 is opposite. Hence the monotonicity is also opposite. But the proof there is still valid for our problem with little modification, here we omit it.
Remark 9. We note that the gradient blowup may occur in infinite time. For example, when the stationary solution v ∈ C 1 (Ω), Theorem 1.3 implies that gradient blowup will occur in infinite time.
4. Proof of the results on finite time gradient blowup. In this section, we will first show that for some large initial data, ∇u must blow up in a finite time. In [17,18], the results were established in one-dimensional and radial cases respectively by constructing self-similar subsolutions. Here, we give a more general subsolution. We refer the readers to [12] for the results on finite time gradient blowup of the equations with generalized nonlinearities. We can also verify that the self-similar subsolution used in [18] is available for our problem. Indeed, we can consider a ball B R ⊂ Ω which is internally tangent to ∂Ω at some point x 0 ∈ ∂Ω. Then applying the same manner as [18,Theorem 2.1] in B R , we can derive lim t→T ∂u ∂ n (x 0 , t) = −∞ for some T < ∞ by which the conclusion follows.
The following local boundary control property for the gradient of the solution v of (15) below will be useful. Its proof can be found in [8, Lemma 2.1].
with maximal existence time T . Suppose that the solution of (1) exists globally in time, i.e. T * = ∞. We can see from [4,8,10,12,20] that ∇v will blow up in finite time on the boundary of Ω. Thus we can set 0 < T < T < ∞. Applying Lemma 2.1 and the fact that 1 6 |∇v| 3 ≤ e |∇v| , we know that u ≥ v in Ω × (0, T ]. Moreover, we have ∂u ∂ n ≤ ∂v ∂ n < 0, on ∂Ω. Then letting T → T and applying Lemma 4.1, we can deduce that ∂u ∂ n ≤ ∂v ∂ n < −∞ at some point x 0 ∈ ∂Ω satisfying lim t→T |∇u(x 0 , t)| = ∞. This contradicts with the assumption that T * = ∞. Hence, ∇u must blow up in a finite time if u 0 is large enough.
Remark 10. From the proof of Theorem 1.4, we can also see that T * ≤ T . The subsolution used above is not unique, we can also construct other subsolutions by using the fact that e x ≥ 1 m! x m for m ∈ Z + , m > 2. Next, we will give a simple proof of Theorem 1.5.
Proof of Theorem 1.5. Here, we just need to show that the solution of problem (15) will have gradient blowup phenomenon.
Let k = 3 and F (t) = 1 k+1 Ω v k+1 dx, then we have By Hölder's inequality, for any ∈ (0, 1 12 ), we have Using Poincaré's inequality, we have Combining (16) with estimates (17) and (18), using Hölder's inequality, we get which is equivalent to Using also the fact that we can conclude that ∇v must blow up in finite time if |Ω| < M is small. Then, following the same procedure as in the proof of Theorem 1.4, we know that ∇u must blow up in finite time.
Remark 11. Following the proof of Theorem 1.5, we can extend this result to the following problem      u t − ∆u = |∇u| p , x ∈ Ω, t > 0, where p > 2.