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 C1 norm to the unique steady state. The main difficulty is to prove C1 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:ut=uxx+eux,0<x<1,t>0,u(0,t)=0,u(1,t)=A,t>0,u(x,0)=u0(x),0<x<1.
Here A>0 is a constant, and the initial data u0 belongs to the space X={v∈C1([0,1]);v(0)=0,v(1)=A} with the C1 norm. The problem (1.1) admits a unique maximum classical solution u=u(u0;·,t), whose existence time will be denoted by T=T*(u0)∈(0,∞]. Note that we make no restriction on the signs of u or ux.

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]u0≤u(x,t)≤max[0,1]u0,0≤x≤1,0≤t<T*.
Since problem (1.1) is well posed in C1, therefore, only three possibilities can occur as follows.

u exists globally and is bounded in C1:
T*=∞supt≥0‖ux(⋅,t)‖∞<∞.
Moreover, due to the results in [3] (see the last part of this Introduction section for more details), u has to converge in C1 to a steady state (which is actually unique when it exists).

u blows up in finite time in C1 norm (finite time gradient blowup):
T*<∞limt→T*‖ux(⋅,t)‖∞=∞.

u exists globally but is unbounded in C1 (infinite time gradient blowup):

T*=∞limsupt→∞‖ux(⋅,t)‖∞=∞.

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 ut=uxx+|ux|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 Ac=1
such that (1.1) has a unique steady-state VA if A<Ac and no steady state if A>Ac (the explicit formula for VA is recalled at the beginning of Section 2). In the critical case A=Ac, there still exists a steady-state VAc, but it is singular, satisfying VAc∈C([0,1])∩C2((0,1]) with VAc,x(0)=∞.

Theorem 1.1.

Assume 0<A<Ac. Then all global solutions of (1.1) are bounded in C1. In other words, (3) cannot occur. Moreover, they converge in C1 norm to VA.

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:

u0,n→u0 in C1,

T*(u0,n)=∞ for each n, and T*(u0)<∞,

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:
L(u(t))=∫01ϕ(u(x,t),ux(x,t))dx.
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 C1, 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 C1 unbounded global solution would exist, we analyze its possible final singularities (along a sequence tn→∞). We shall show that ux remains bounded away from the left boundary and describe the shape of ux 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 C1, 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 Wx(0)=∞. But such a W is not available if A≠Ac, leading to a contradiction.

2. 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 t0∈(0,T*), there exists C1>0 such that
|ut|≤C1,0≤x≤1,t0≤t<T*.

Proof.

The function h=ut satisfies
ht=hxx+euxhx,0<x<1,t0<t<T*,h(0,t)=h(1,t)=0,t0<t<T*,h(x,t0)=uxx(x,t0)+eux(t0,x),0<x<1.
It follows from the maximum principle that |h|≤∥h(t0)∥∞ in [0,1]×[t0,T*).

Remark 2.2.

Although the second-order compatibility condition is not assumed, the maximum principle is still valid for ut. 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 u0∈H2∩H01, standard regularity results imply ut∈C([t0,T);L2), which is enough to apply the (weak) Stampacchia maximum principle to the function h=ut (which satisfies ht=hxx+a(x,t)hx with a(x,t) bounded near t=t0).

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

Lemma 2.3.

Let u be a maximal solution of (1.1). For all t0∈(0,T*), there exists C1>0 such that, for all 0≤x≤1 and t0≤t<T*,
ux(x,t)≤C1x+ln1x+e-ux(0,t),ux(1-x,t)≥-C1x-ln1x+e-ux(0,t).

Proof.

Fix t∈[t0,T*) and let y(x)=(ux(x,t)-C1x)+, where C1 is given by Lemma 2.1. The function y satisfies
y′+ey=(uxx-C1)|{ux>C1x}+e(ux-C1x)+.
For each x such that ux(x,t)>C1x, we have y′+ey≤uxx-C1+eux≤0 by Lemma 2.1. Therefore, we have y′+ey≤0 on (0,1). By integration, it follows that y(x)≤ln1/[x+e-ux(0,t)], hence, (2.3).

As for (2.4), it follows similarly by considering y(x)=(-ux(1-x,t)-C1x)+.

Lemma 2.4.

Let u be a maximal solution of (1.1). There exists C2>0 such that, for all T∈(0,T*),
maxQTux(x,t)≤max(C2,max0≤t≤Tux(0.t)),
where QT=[0,1]×[0,T] and
minQTux(x,t)≥min(-C2,min0≤t≤Tux(1.t)).

Proof.

The function w=ux satisfies wt=wxx+a(x,t)wx in (0,1)×(0,T*), where a(x,t)=eux. Therefore, w attains its extrema in QT on the parabolic boundary of QT.

Since, by Lemma 2.3, we have ux(1,t)≤C and ux(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 ux in case that ux(1,t) or ux(0,t) becomes unbounded.

Lemma 2.5.

Let u be a maximal solution of (1.1). For all t0∈(0,T*), there exists C3>0 such that, for all 0≤x≤1 and t0≤t<T*,
e-[ux+(x,t)+C3]≤e-[ux+(0,t)+C3]+x,e-[(-ux)+(1-x,t)+C3]≤e-[(-ux)+(1,t)+C3]+x.

Proof.

Fix t∈[t0,T*), and let z(x)=ux+(x,t)+ln(1+C1), where C1 is given by Lemma 2.1. The function z satisfies
z′+ez=uxx|{ux>0}+eux+(x,t)+ln(1+C1)≥(uxx+eux+C1eux)|{ux>0}≥(uxx+eux+C1)|{ux>0}≥0,
on [0,1] by Lemma 2.1. By integration, it follows that e-z(x)≤e-z(0)+x, that is, (2.8) with C3=ln(1+C1).

The estimate (2.9) follows similarly by considering Z(x)=(-ux)+(1-x,t)+ln(1+C1).

Lemma 2.6.

Let u be a global solution of (1.1). Then it holds
inf[0,1]×[0,∞)ux>-∞.

Proof.

Assume that the lemma is false. Then, by Lemma 2.4, there exists a sequence tn→∞ such that ux(1,tn)→-∞.

Fix ɛ>0. By (2.9) in Lemma 2.5, for n>n0(ɛ) large enough, we have
e-[(-ux)+(1-x,tn)+ln(1+C1)]≤e-[(-ux)+(1,tn)+ln(1+C1)]+x≤ɛ,0≤x≤ɛ.
Hence,
(-ux)+(1-x,tn)≥-lnɛ-ln(1+C1),0≤x≤ɛ.
By choosing ɛ=ɛ(C1) small, we deduce that ux(1-x,tn)≤-1 on [0,ɛ]; hence,
u(1-x,tn)≥A+x,0≤x≤ɛ,
for all n≥n0(ɛ). But this contradicts the strong maximum principle which implies that limt→∞{maxx∈[0,1]u(x,t)}≤A.

3. Lyapunov’s Functional and Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>

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 C1.

Proposition 3.1.

Fix any K>0 and let DK=[-K,K]×ℝ. There exist functions ϕ∈C1(DK;ℝ) and ψ∈C(DK;(0,∞)) with the following property: for any solution u of (1.1) with |u|≤K, defining
L(u(t)):=∫01ϕ(u(x,t),ux(x,t))dx,
it holds
ddtL(u(t))=-∫01ψ(u(x,t),ux(x,t))ut2(x,t)dx,0<t<T*.
Furthermore, we have
ϕ≥0.

Proof.

For a given function φ(u,v), let us denote
H=φu+evφvv-vφuv.
Here we assume that φ, φu, φv, andφvu are continuous and C1 in v in DK and that φvv is continuous in DK. We observe that H is continuous and differentiable in v in DK and satisfies
Hv=evφvvv+evφvv-vφuvv.
Now suppose that ψ:=φvv satisfies
vψu-evψv-evψ=0,|u|≤K.
It follows that Hv=0; hence,
H=H(u)=φu(u,0).
Let then
ϕ(u,v)=φ(u,v)-∫0uH(s)ds=φ(u,v)-φ(u,0)+φ(0,0).
We compute, using integration by parts and ut(1,t)=0 and ut(0,t)=0,
ddtL(u(t))=∫01{(φu(u,ux)-H(u))ut+φv(u,ux)uxt}(x,t)dx=∫01{(φu(u,ux)-H(u)-φvu(u,ux)ux-φvv(u,ux)uxx)}ut(x,t)dx.
Using the definition of H and uxx=ut-eux, we deduce that
ddtL(u(t))=-∫01ψ(u(x,t),ux(x,t))ut2(x,t)dx.
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
ψ(u,v)=ev>0
is a solution of (3.6) on [-K,K]×ℝ.

Define φ by
φ(u,v)=∫0v∫0zψ(u,s)dsdz≥0.
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])∩C2(0,1] of
Wxx+eWx=0,0<x<1,W(0)=0,W(1)=A.
Moreover, the convergence also holds in C1([ɛ,1]) for all ɛ>0.

Proof.

Fix any sequence tn→∞, and let un=u(·,tn+·). Denote Q:=[0,1]×[0,∞) and Qɛ:=(ɛ,1]×[0,∞),for all ɛ>0.

From (1.2) and Lemma 2.1, we know that
|u|+|ut|≤Cin[0,1]×[1,∞).
Also, using (2.3) and Lemma 2.6, we obtain
‖∂xun‖L∞(1,∞;L∞(0,1))≤C.
It follows from (3.14) and (3.15) that the sequence {(un)} 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 |ux|≤C(ɛ), and; hence, |uxx|≤C(ɛ) in (ɛ,1]×[1,∞). Since w:=ux satisfies wt-wxx=euxwx, parabolic regularity estimates then imply that
‖wt(⋅,tn+⋅)‖L∞((ɛ,1]×(0,T))≤C(ɛ,T),T>0.
It follows that the sequence {∂xun} is relatively compact in C([ɛ,1]×[0,T]) for each ɛ,T>0. Then some subsequence {unk} converges to a function W∈C(Q¯), with wx∈C(Q), which satisfies
Wt-Wxx=eWxinQ,W(0,t)=0,W(1,t)=A,t≥0.
The convergence of {unk} is uniform in each set [0,1]×[0,T], and the convergence of {∂xunk} is uniform in each set [ɛ,1]×[0,T].

Now, by (1.2), we may find K>0 such that
|u|≤Kon[0,1]×[0,∞).
Since ψ, given by Proposition 3.1, is positive and continuous, we have
η(K,R):=inf{ψ(u,v);|u|≤K,|v|≤R}>0,∀R>0.
Fix any ɛ∈(0,1). We get, for all T>1,
η(K,C(ɛ))∫1T∫ɛ1ut2(x,t)dxdt≤∫1T∫01ψ(u,ux)ut2(x,t)dxdt=L(u(1))-L(u(T))≤L(u(1)).
This implies that ∫1∞∫ɛ1ut2(x,t)dxdt<∞; hence,
∫0∞∫ɛ1(∂tunk2)(x,t)dxdt=∫tnk∞∫ɛ1ut2(x,t)dxdt→0,k→∞.
Since ∂tunk→Wt in 𝒟′((0,1)×(0,∞)) and since ɛ∈(0,1) is arbitrary, it follows that Wt≡0. Therefore, W=W(x)∈C([0,1])∩C2(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 tn→∞ was arbitrary, this readily implies that the whole solution u(t) actually converges to W. The proposition is proved.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

For 0<A<Ac, assume that u is a global solution of (1.1) which is unbounded in C1. By Proposition 3.2, as t→∞, u(t) converges to W=VA, with convergence in C([0,1]) and in C1([ɛ,1]) for all ɛ>0.

Since u is unbounded, by Lemmas 2.4 and 2.6, there exists a sequence tn→∞ such thatux(0,tn)→∞.
Using Lemma 2.6, (2.8), and (3.22), we deduce that Wx(x)≥-C and
e-[Wx+(x)]+C3≤xin(0,1].
This easily implies that
Wx(x)≥-lnx-C′in(0,1].
But this is a contradiction, since W=VA∈C([0,1]). We have, thus, proved that all global solutions are bounded in C1.

Finally, once boundedness is known, the convergence of global solutions to VA in C1 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 <xref ref-type="statement" rid="prop1.1">1.3</xref>.

Let
D={u0∈X;u(u0;⋅,t)convergestoVAinC1ast→∞}
and fix A¯∈(A,Ac). We claim that,
∀u0∈X,u0≤min(A,VA¯)impliesu0∈D.
Indeed, by the comparison principle, as long as u:=u(u0;·,t) exists, we have u≤VA¯; hence, ux(0,t)≤VA¯,x(0), and u≤A; hence, ux(1,t)≥0. By Lemma 2.4, we deduce that u is global and bounded in C1. It then follows from [3] that u converges in C1 to the unique steady-state VA as t→∞, which proves the claim.

Let us first consider the case A∈(0,Ac). By [4], there exists u0¯∈X with u¯0,x≥0, such that T*(u¯0)<∞. For each λ∈[0,1], denote u0,λ:=VA+λ(u¯0-VA)∈X and uλ:=u(u0,λ;·,t). For λ>0 small, we have u0,λ≤min(A,VA¯); hence,u0,λ∈D. Therefore, λ*:=inf{λ∈[0,1];u0,λ∉D}∈(0,1]. By (3.26) and a standard continuous dependence argument, we have u0,λ*∉D. This implies that uλ* cannot be global and bounded in C1 (since otherwise it would converge to VA due to [3]). In view of Theorem 1.1, the only remaining possibility is that T*(u0,λ*)<∞. Considering u0,λn for a sequence λn↑λ*, we obtain the conclusions (a) and (b) of Proposition 1.3. We also get (c), since otherwise uλ* would be global by continuous dependence.

Acknowledgments

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.

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