We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded or the solution is global and converges in norm to the unique steady state. The main difficulty is to prove boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov's functional by carrying out the method of Zelenyak.

1. Introduction and Main Results

We consider the problem: Here is a constant, and the initial data belongs to the space with the norm. The problem (1.1) admits a unique maximum classical solution , whose existence time will be denoted by . Note that we make no restriction on the signs of or .

The differential equation in (1.1) possesses both mathematical and physical interest. It can serve as a typical model case in the theory of parabolic PDEs. Indeed, it is the one of the simplest examples (along with Burger's equation) of a parabolic equation with a nonlinearity depending on the first-order spatial derivatives of . On the other hand, this equation (and its -dimensional version) arises in the viscosity approximation of the Hamilton-Jacobi-type equations from stochastic control theory [1] and in some physical models of surface growth [2].

The aim of this paper is to provide a complete classification of large time behavior of the solutions of (1.1). A basic fact about (1.1) is that the solutions satisfy a maximum principle: Since problem (1.1) is well posed in , therefore, only three possibilities can occur as follows.(1) exists globally and is bounded in : Moreover, due to the results in [3] (see the last part of this Introduction section for more details), has to converge in to a steady state (which is actually unique when it exists).(2) blows up in finite time in norm (finite time gradient blowup): (3) exists globally but is unbounded in (infinite time gradient blowup):

In [4], the first author and Hu studied the case (2) and got estimates on the gradient blowup rate under the assumptions on the initial data so that the solution is monotone in and in . In the present paper, our primary goal is to exclude (3), that is, infinite time gradient blowup. For the boundedness of global solutions of other problems, for example, the equation with , we refer to [5] and the references therein.

For , the situation is slightly more involved. There exists a critical value such that (1.1) has a unique steady-state if and no steady state if (the explicit formula for is recalled at the beginning of Section 2). In the critical case , there still exists a steady-state , but it is singular, satisfying with .

Theorem 1.1. Assume . Then all global solutions of (1.1) are bounded in . In other words, (3) cannot occur. Moreover, they converge in norm to .

For the case , we improve the result by removing the restrictions and on the initial data. Then all solutions of (1.1) blow up in finite time in norm.

Remark 1.2. In the critical case , all solutions have to blow up in in either finite or infinite time. Moreover, if (3) occurs, then the solution will converge in to the singular steady-state , as . This follows from Proposition 3.2 below. However, the possibility of (3) remains an open problem in this case. We conjecture that this could occur.
As a consequence of our results, we exhibit the following interesting situation: although boundedness of global solutions is true, the global solutions of (1.1) do not satisfy a uniform a priori estimate, that is, the supremum in (1) cannot be estimated in terms of the norm of the initial data. In other words, there exists a bounded, even compact, subset , such that the trajectories starting from describe an unbounded subset of , although each of them is individually bounded and converges to the same limit. As a further consequence, the existence time , defined as a function from into , is not (upper semi) continuous.

Proposition 1.3. Assume . There exists and a sequence in with the following properties:(a) in ,(b) for each , and ,(c).

To explain the ideas of our proof, let us first recall that, in a classical paper [3], Zelenyak showed that any one-dimensional quasilinear uniformly parabolic equation possesses a (strict) Lyapunov’s functional, of the form: The construction of is in principle explicit, although too complicated to be completely computed in most situations. As a consequence, for any solution of (1.1) which is global and bounded in , the (nonempty) -limit set of consists of equilibria. Since (1.1) admits at most one equilibrium , such has to converge to . (In fact, it was also proved in [3] that whether or not equilibria are unique, any bounded solution of a one-dimensional uniformly parabolic equation converges to an equilibrium, but this need not concern us here.) For , our proof proceeds by contradiction and makes essential use of the Zelenyak construction. It consists of three steps as follows.

Assuming that a unbounded global solution would exist, we analyze its possible final singularities (along a sequence ). We shall show that remains bounded away from the left boundary and describe the shape of near the boundary (cf. Section 2).

We shall carry out the Zelenyak construction in a sufficiently precise way to determine the density of the Lyapunov functional. It will turn out that, whenever remains in a bounded set of (as it does here in view of the estimate (1.2)), remains bounded from below uniformly with respect to (see Proposition 3.1).

Using this property of in the classical Lyapunov’s argument, together with the fact that singularities may occur only near the boundary, it will be possible to prove the following convergence result: any global solution, even unbounded in , has to converge in to a stationary solution of (1.1) with , (see Proposition 3.2). On the other hand, if were bounded, then our estimates would imply . But such a is not available if , leading to a contradiction.

2. Preliminary Estimates

We start with some preliminary estimates. They are collected in Lemmas 2.12.6.

Lemma 2.1. Let be a maximal solution of (1.1). For all , there exists such that

Proof. The function satisfies It follows from the maximum principle that in .

Remark 2.2. Although the second-order compatibility condition is not assumed, the maximum principle is still valid for . In fact, the system can be approximated by boundary data satisfying the second-order compatibility condition and taking the limit, or another simpler argument (without approximation procedure) is this: since , standard regularity results imply , which is enough to apply the (weak) Stampacchia maximum principle to the function (which satisfies with bounded near ).

The following two lemmas give upper and lower bounds on which show, in particular, that remains bounded away from the boundary.

Lemma 2.3. Let be a maximal solution of (1.1). For all , there exists such that, for all and ,

Proof. Fix and let , where is given by Lemma 2.1. The function satisfies For each such that , we have by Lemma 2.1. Therefore, we have on . By integration, it follows that hence, (2.3).
As for (2.4), it follows similarly by considering .

Lemma 2.4. Let be a maximal solution of (1.1). There exists such that, for all , where and
Proof. The function satisfies in , where . Therefore, attains its extrema in on the parabolic boundary of .
Since, by Lemma 2.3, we have and for all , the conclusion follows.

The following lemma will provide a useful lower bound on the blowup profile of in case that or becomes unbounded.

Lemma 2.5. Let be a maximal solution of (1.1). For all , there exists such that, for all and ,

Proof. Fix , and let , where is given by Lemma 2.1. The function satisfies on by Lemma 2.1. By integration, it follows that , that is, (2.8) with .
The estimate (2.9) follows similarly by considering .

Lemma 2.6. Let be a global solution of (1.1). Then it holds

Proof. Assume that the lemma is false. Then, by Lemma 2.4, there exists a sequence such that .
Fix . By (2.9) in Lemma 2.5, for large enough, we have Hence, By choosing small, we deduce that on ; hence, for all . But this contradicts the strong maximum principle which implies that .

3. Lyapunov’s Functional and Proof of Theorem 1.1

As a main step, we now carry out the argument of Zelenyak to construct a Lyapunov’s functional. The key point here is that the Lyapunov functional enjoys nice properties on any global trajectory of (1.1), even if it were unbounded in .

Proposition 3.1. Fix any and let . There exist functions and with the following property: for any solution of (1.1) with , defining it holds Furthermore, we have
Proof. For a given function , let us denote Here we assume that , , , and are continuous and in in and that is continuous in . We observe that is continuous and differentiable in in and satisfies Now suppose that satisfies It follows that ; hence, Let then We compute, using integration by parts and and , Using the definition of and , we deduce that We have, thus, obtained (3.2), provided (3.6) is true.
Now, (3.6) can be solved by the method of characteristics. For each , one finds that the function defined by is a solution of (3.6) on .
Define by It is easy to check that enjoys the regularity properties assumed at the beginning of the proof and ; hence, .

As a consequence of Proposition 3.1 and of Section 2, we shall obtain the following convergence result. Of course, the main point here is that we do not assume to be bounded, but only global.

Proposition 3.2. Let be a global solution of (1.1). Then, as , converges in to a stationary solution of (1.1), that is, a function of Moreover, the convergence also holds in for all .

Proof. Fix any sequence , and let . Denote and for all .
From (1.2) and Lemma 2.1, we know that Also, using (2.3) and Lemma 2.6, we obtain It follows from (3.14) and (3.15) that the sequence is relatively compact in for each .
On the other hand, using (2.3), (2.4), and (3.14), we have , and; hence, in . Since satisfies , parabolic regularity estimates then imply that It follows that the sequence is relatively compact in for each . Then some subsequence converges to a function , with , which satisfies The convergence of is uniform in each set , and the convergence of is uniform in each set .
Now, by (1.2), we may find such that Since , given by Proposition 3.1, is positive and continuous, we have Fix any . We get, for all , This implies that ; hence, Since in and since is arbitrary, it follows that . Therefore, satisfies (3.13).
But we know (cf. the beginning of Section 2) that the solution of (3.13) is unique whenever it exists. Since the sequence was arbitrary, this readily implies that the whole solution actually converges to . The proposition is proved.

Proof of Theorem 1.1. For , assume that is a global solution of (1.1) which is unbounded in . By Proposition 3.2, as , converges to , with convergence in and in for all .
Since is unbounded, by Lemmas 2.4 and 2.6, there exists a sequence such that Using Lemma 2.6, (2.8), and (3.22), we deduce that and This easily implies that But this is a contradiction, since . We have, thus, proved that all global solutions are bounded in .
Finally, once boundedness is known, the convergence of global solutions to in is a standard consequence of the existence of a Lyapunov’s functional, the uniqueness of the steady-state, and compactness properties of the semi-flow associated with (1.1). The proof of Theorem 1.1 is completed.

Proof of Proposition 1.3. Let and fix . We claim that, Indeed, by the comparison principle, as long as exists, we have ; hence, , and ; hence, . By Lemma 2.4, we deduce that is global and bounded in . It then follows from [3] that converges in to the unique steady-state as , which proves the claim.
Let us first consider the case . By [4], there exists with , such that . For each , denote and . For small, we have ; hence,. Therefore, . By (3.26) and a standard continuous dependence argument, we have . This implies that cannot be global and bounded in (since otherwise it would converge to due to [3]). In view of Theorem  1.1, the only remaining possibility is that . Considering for a sequence , we obtain the conclusions and of Proposition 1.3. We also get , since otherwise would be global by continuous dependence.


The authors would thank the anonymous referee very much for his valuable corrections and suggestions. This work was supported by the Fundamental Research Funds for the Central Universities of China.