Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source

and Applied Analysis 3 For the case A > Ac, we improve the result by removing the restrictions u0 ≥ 0 and u0 x ≥ 0 on the initial data. Then all solutions of 1.1 blow up in finite time in C1 norm. Remark 1.2. In the critical case A Ac, all solutions have to blow up in C1 in either finite or infinite time. Moreover, if 3 occurs, then the solution will converge in C 0, 1 to the singular steady-state VAc , as t → ∞. 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 C1 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 S ⊂ X, such that the trajectories starting from S describe an unbounded subset of X, although each of them is individually bounded and converges to the same limit. As a further consequence, the existence time T ∗, defined as a function from X into 0,∞ , is not upper semi continuous. Proposition 1.3. Assume 0 < A < Ac. There exists u0 ∈ X and a sequence {u0,n} in X with the following properties: a u0,n → u0 in C1, b T ∗ u0,n ∞ for each n, and T ∗ u0 <∞, c supt≥0‖ un x ·, t ‖∞ : Kn → ∞. 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:


Introduction and Main Results
We consider the problem:

1.1
Here A > 0 is a constant, and the initial data u 0 belongs to the space X {v ∈ C 1 0, 1 ; v 0 0, v 1 A} with the C 1 norm.The problem 1.1 admits a unique maximum classical solution u u u 0 ; •, t , whose existence time will be denoted by T T * u 0 ∈ 0, ∞ .Note that we make no restriction on the signs of u or u x .
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 u.On the other hand, this equation and its N-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: min 0,1 Since problem 1.1 is well posed in C 1 , therefore, only three possibilities can occur as follows.
1 u exists globally and is bounded in C 1 : Moreover, due to the results in 3 see the last part of this Introduction section for more details , u has to converge in C 1 to a steady state which is actually unique when it exists .
2 u blows up in finite time in C 1 norm finite time gradient blowup : 3 u exists globally but is unbounded in C 1 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 x and in t.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 u t u xx |u x | p with p > 2, we refer to 5 and the references therein.
For A > 0, the situation is slightly more involved.There exists a critical value such that 1.1 has a unique steady-state V A if A < A c and no steady state if A > A c the explicit formula for V A is recalled at the beginning of Section 2 .In the critical case A A c , there still exists a steady-state Theorem 1.1.Assume 0 < A < A c .Then all global solutions of 1.1 are bounded in C 1 .In other words, (3) cannot occur.Moreover, they converge in C 1 norm to V A .
For the case A > A c , we improve the result by removing the restrictions u 0 ≥ 0 and u 0 x ≥ 0 on the initial data.Then all solutions of 1.1 blow up in finite time in C 1 norm.

Remark 1.2. In the critical case A
A c , all solutions have to blow up in C 1 in either finite or infinite time.Moreover, if 3 occurs, then the solution will converge in C 0, 1 to the singular steady-state V A c , as t → ∞.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 C 1 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 S ⊂ X, such that the trajectories starting from S describe an unbounded subset of X, although each of them is individually bounded and converges to the same limit.As a further consequence, the existence time T * , defined as a function from X into 0, ∞ , is not upper semi continuous.

Proposition 1.3. Assume 0 < A < A c
. There exists u 0 ∈ X and a sequence {u 0,n } in X with the following properties: ∞ for each n, and 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 φ is in principle explicit, although too complicated to be completely computed in most situations.As a consequence, for any solution u of 1.1 which is global and bounded in C 1 , the nonempty w-limit set of u consists of equilibria.Since 1.1 admits at most one equilibrium V , such u has to converge to V .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 A > 0, our proof proceeds by contradiction and makes essential use of the Zelenyak construction.It consists of three steps as follows.
Assuming that a C 1 unbounded global solution would exist, we analyze its possible final singularities along a sequence t n → ∞ .We shall show that u x remains bounded away from the left boundary and describe the shape of u x near the boundary cf.Section 2 .
We shall carry out the Zelenyak construction in a sufficiently precise way to determine the density φ u, v of the Lyapunov functional.It will turn out that, whenever u remains in a bounded set of R as it does here in view of the estimate 1.2 , φ u, v remains bounded from below uniformly with respect to v see Proposition 3.1 .
Using this property of φ 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 C 1 , has to converge in C 0, 1 to a stationary solution W of 1.1 with W 0 0, W 1 A see Proposition 3.2 .On the other hand, if u were bounded, then our estimates would imply W x 0 ∞.But such a W is not available if A / A c , leading to a contradiction.

Preliminary Estimates
We start with some preliminary estimates.They are collected in Lemmas 2.1-2.6.Lemma 2.1.Let u be a maximal solution of 1.1 .For all t 0 ∈ 0, T * , there exists C 1 > 0 such that 2.1
Remark 2.2.Although the second-order compatibility condition is not assumed, the maximum principle is still valid for u t .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 u 0 ∈ H 2 ∩ H 1 0 , standard regularity results imply u t ∈ C t 0 , T ; L 2 , which is enough to apply the weak Stampacchia maximum principle to the function h u t which satisfies h t h xx a x, t h x with a x, t bounded near t t 0 .
The following two lemmas give upper and lower bounds on u x which show, in particular, that u x remains bounded away from the boundary.Lemma 2.3.Let u be a maximal solution of 1.1 .For all t 0 ∈ 0, T * , there exists C 1 > 0 such that, for all 0 ≤ x ≤ 1 and t 0 ≤ t < T * , Proof.Fix t ∈ t 0 , T * and let y x u x x, t − C 1 x , where C 1 is given by Lemma 2.1.The function y satisfies y e y u xx − C 1 | {u x >C 1 x} e u x −C 1 x .

2.5
For each x such that u x x, t > C 1 x, we have y e y ≤ u xx − C 1 e u x ≤ 0 by Lemma 2.1.Therefore, we have y e y ≤ 0 on 0, 1 .By integration, it follows that y x ≤ ln 1/ x e −u x 0,t , hence, 2.3 .As for 2.4 , it follows similarly by considering y x −u x 1 − x, t − C 1 x .
Lemma 2.4.Let u be a maximal solution of 1.1 .There exists C 2 > 0 such that, for all T ∈ 0, T * , max where Q T 0, 1 × 0, T and min Proof.The function w u x satisfies w t w xx a x, t w x in 0, 1 × 0, T * , where a x, t e u x .Therefore, w attains its extrema in Q T on the parabolic boundary of Q T .
Since, by Lemma 2.3, we have u x 1, t ≤ C and u x 0, t ≥ −C for all t ∈ 0, T * , the conclusion follows.
The following lemma will provide a useful lower bound on the blowup profile of u x in case that u x 1, t or u x 0, t becomes unbounded.Lemma 2.5.Let u be a maximal solution of 1.1 .For all t 0 ∈ 0, T * , there exists C 3 > 0 such that, for all 0 ≤ x ≤ 1 and t 0 ≤ t < T * , e − u x x,t C 3 ≤ e − u x 0,t C 3 x, 2.8 e − −u x 1−x,t C 3 ≤ e − −u x 1,t C 3 x.

2.9
Proof.Fix t ∈ t 0 , T * , and let z x u x x, t ln 1 C 1 , where C 1 is given by Lemma 2.1.The function z satisfies 2.10 on 0, 1 by Lemma 2.1.By integration, it follows that e −z x ≤ e −z 0 x, that is, 2.8 with The estimate 2.9 follows similarly by considering

2.11
Proof.Assume that the lemma is false.Then, by Lemma 2.4, there exists a sequence t n → ∞ such that u x 1, t n → −∞.
Fix ε > 0. By 2.9 in Lemma 2.5, for n > n 0 ε large enough, we have By choosing ε ε C 1 small, we deduce that u x 1 − x, t n ≤ −1 on 0, ε ; hence, for all n ≥ n 0 ε .But this contradicts the strong maximum principle which implies that lim t → ∞ {max x∈ 0,1 u x, t } ≤ A.

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 C 1 .
Proposition 3.1.Fix any K > 0 and let D K −K, K × R.There exist functions φ ∈ C 1 D K ; R and ψ ∈ C D K ; 0, ∞ with the following property: for any solution u of 1.1 with |u| ≤ K, defining Furthermore, we have Proof.For a given function ϕ u, v , let us denote H ϕ u e v ϕ vv − vϕ uv .

3.4
Here we assume that ϕ, ϕ u , ϕ v , and ϕ vu are continuous and C 1 in v in D K and that ϕ vv is continuous in D K .We observe that H is continuous and differentiable in v in D K and satisfies H v e v ϕ vvv e v ϕ vv − vϕ uvv .

3.6
It follows that H v 0; hence, We compute, using integration by parts and u t 1, t 0 and u t 0, t 0,

3.9
Using the definition of H and u xx u t − e u x , we deduce that ψ u x, t , u x x, t u 2 t x, t dx.

3.10
We have, thus, obtained 3.2 , provided 3.6 is true.Now, 3.6 can be solved by the method of characteristics.For each K > 0, one finds that the function ψ defined by

3.12
It is easy to check that ϕ enjoys the regularity properties assumed at the beginning of the proof and φ ϕ; hence, φ ≥ 0.
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 u to be bounded, but only global.Proposition 3.2.Let u be a global solution of 1.1 .Then, as t → ∞, u t converges in C 0, 1 to a stationary solution of 1.1 , that is, a function W ∈ C 0, 1 ∩ C 2 0, 1 of
Proof.Fix any sequence t n → ∞, and let From 1.2 and Lemma 2.1, we know that Also, using 2.3 and Lemma 2.6, we obtain

3.15
It follows from 3.14 and 3.15 that the sequence { u n } is relatively compact in C 0, 1 × 0, T for each T > 0.
On the other hand, using 2.3 , 2.4 , and 3.14 , we have |u x | ≤ C ε , and; hence, Since w : u x satisfies w t − w xx e u x w x , parabolic regularity estimates then imply that

3.17
The convergence of {u n k } is uniform in each set 0, 1 × 0, T , and the convergence of {∂ x u n k } is uniform in each set ε, 1 × 0, T .Now, by 1.2 , we may find K > 0 such that |u| ≤ K on 0, 1 × 0, ∞ .

3.20
This implies that

3.21
Since ∂ t u n k → W t in D 0, 1 × 0, ∞ and since ε ∈ 0, 1 is arbitrary, it follows that W t ≡ 0. Therefore, W W x ∈ C 0, 1 ∩ C 2 0, 1 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 t n → ∞ was arbitrary, this readily implies that the whole solution u t actually converges to W. The proposition is proved.
Proof of Theorem 1.1.For 0 < A < A c , assume that u is a global solution of 1.1 which is unbounded in C 1 .By Proposition 3.2, as t → ∞, u t converges to W V A , with convergence in C 0, 1 and in C 1 ε, 1 for all ε > 0.
Since u is unbounded, by Lemmas 2.4 and 2.6, there exists a sequence t n → ∞ such that u x 0, t n → ∞.

3.22
Using Lemma 2.6, 2.8 , and 3.22 , we deduce that W x x ≥ −C and e − W x x C 3 ≤ x in 0, 1 .

3.23
This easily implies that W x x ≥ − ln x − C in 0, 1 .

3.24
But this is a contradiction, since W V A ∈ C 0, 1 .We have, thus, proved that all global solutions are bounded in C 1 .
Finally, once boundedness is known, the convergence of global solutions to V A in C 1 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.