STABILIZATION TOWARDS A SINGULAR STEADY STATE WITH GRADIENT BLOW-UP FOR A DIFFUSION-CONVECTION PROBLEM

This paper is devoted to analyze a case of singularity formation 
in infinite time for a semilinear heat equation involving linear 
diffusion and superlinear convection. A feature to be noted is 
that blow-up happens not for the main unknown but for its 
derivative. The singularity builds up at the boundary. The 
formation of inner and outer regions is examined, as well as the 
matching between them. As a consequence, we obtain the precise 
exponential rates of blow-up in infinite time.


Introduction and main results
In this paper we perform a blow-up analysis for the problem (1.1) We take parameters p > 2, M ≥ 0, and the initial data u 0 belong to the space X = {v ∈ C 1 ([0, 1]); v(0) = 0, v(1) = M }, endowed with the C 1 norm.Problem (1.1) admits a unique maximal classical solution u = u(t, x), whose existence time will be denoted by The differential equation in (1.1) possesses both mathematical and physical interest.This equation (and its N -dimensional version) arises in the viscosity approximation of Hamilton-Jacobi type equations from stochastic control theory [21] and in some physical models of surface growth [18].
On the other hand, 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 Burgers' equation) of a parabolic equation with a nonlinearity depending on the first-order spatial derivatives of u.It can be considered as an analogue of the extensively studied reaction-diffusion equation with zero order nonlinearity u t − u xx = u p .There are some significant differences: while the latter equation is well-known to exhibit L ∞ blow-up whenever p > 1, only gradient blow-up can occur in the former and only for p > 2 (actually, it is well-known that all solutions are global for 0 < p ≤ 2, cf.[19]).By gradient blow-up, we mean that u x blows up in L ∞ norm as t → T * , whereas u remains uniformly bounded (see e.g.[24] and the references therein for details).
It is also interesting to write Problem (1.1) in an equivalent way in terms of the derivative v = u x .We have Here c(v) = p |v| p−2 v.In this formulation, the governing equation combines linear diffusion and superlinear convection.Understanding blow-up phenomena for diffusion-convection equations like (1.2) has an added interest in view of the current effort to solve the much more difficult problems of blow-up for general fluid flows.
Due to the fact that the v-formulation includes nonlinear reaction/absorption on the boundary, it is more natural to perform the asymptotic analysis in terms of u, but we must bear in mind that the quantity that blows up is v = u x .
Stationary states and blow-up.It is a well-known fact that the large-time behavior of evolution equations is closely connected to the existence and properties of the stationary states.In this respect, the following picture holds.For small values of M ≥ 0, Problem (1.1) admits a unique steady state . Moreover, the steady states are ordered, i.e.V M < V M in (0, 1] for 0 ≤ M < M < M c .On the other hand, there is no steady state if M > M c .In the critical case M = M c , there still exists a steady state V Mc = U , given by the explicit formula ), but it is singular in the sense that it has infinite derivative on the left-hand boundary, Next, a basic fact about (1.1) is that the solutions satisfy a maximum principle: (1.4) min Since Problem (1.1) is well-posed in C 1 locally in time, only three possibilities can occur: (I) u exists globally and is bounded in C 1 : (II) u blows up in finite time in C 1 norm (finite time gradient blow-up): (III) u exists globally but is unbounded in C 1 (infinite time gradient blow-up): For M = M c , the global behavior of solutions of (1.1) is rather well understood [1,10,24,3].In particular, it is proved in paper [3] that (III) never occurs.Moreover, if 0 ≤ M < M c , then every solution either blows up in finite time (this occurs for suitably large initial data) or converges to the unique steady state V M (which in turn is stable in C 1 ).On the contrary, if M > M c , then all solutions blow up in finite time.
In the critical case M = M c , all solutions have to blow up in C 1 in either finite or infinite time and all large enough solutions do so in finite time.Moreover, it was proved in [3] that if (III) occurs, then the solution will converge in C([0, 1]) to the singular steady state U , as t → ∞.
However, the possibility of (III) remained an open problem and is established in this paper in the critical case M = M c .
Main result.The situation for "small" solutions in the critical case will turn out to be very different from the subcritical case.Namely, we will show that all solutions with initial data u 0 below the singular steady state U are global in time and exhibit infinite time gradient blow-up.To be precise, our assumptions are: In that situation, the solution u to Problem (1.1) is globally bounded in L ∞ and gradient blowup occurs (only) at the left boundary.More precisely, it is known that in such a case, u will satisfy max [3,Section 2]).Moreover, as a consequence of Lyapunov functional analysis and regularity results, cf.[3, Proposition 3.2], u(t) must converge to U in C([0, 1]) as t → ∞.
A natural question is thus to investigate the rate of divergence of u x (t, 0).We will answer this question, together with a rather precise description of the asymptotic behavior of u(t, x) as t → ∞.The main facts are stated in the following result.
Theorem 1.1 Assume p > 2, M = M c and conditions (1.5).Then the solution u of Problem (1.1) is global in time, it satisfies −C ≤ u(t, x) ≤ U (x) for all x ∈ [0, 1] and all t > 0 and we have the uniform exponential convergence (1.6) lim as well as the grow-up estimate The exponent λ = λ(p) > 0 is defined in Proposition 5.1 and Theorem 5.1 as the first eigenvalue of an associated linearized problem, and µ = λ/(p − 2).
Remark 1.1.Theorem 1.1 seems to be the first example of gradient blow-up in infinite time for a semilinear parabolic equation.For an example in a quasilinear equation, we refer to [9], where the problem of mean curvature type u t = (1 + u 2 x ) −1/2 u x x + λu has been considered.However, no estimate of the grow-up rate is given there.As for L ∞ blow-up in infinite time, examples have been known for some time, e.g. for the equation u t − ∆u = u p (see [22,15,23]) and the grow-up rate has been studied in the paper [11], whose outline of the matching process we follow closely; the method has been applied in [14] to fast diffusion; see also [13] for a different problem with convection and nonlinear boundary conditions.
The rest of the paper is devoted to the detailed study of (1.1) and to the proof of Theorem 1.1.Global existence is proved in Section 2. In Section 3, we give simple proofs of weaker versions of some of the estimates in Theorem 1.1 (exponential convergence in a weighted L 2 norm and an exponential lower bound of u x (t, 0)).In Sections 4 and 5, further description of the asymptotic behavior of u is provided.Namely, the formation of inner (boundary layer) and outer regions is examined.Finally, a matching procedure between the inner and outer regions allows us to conclude the proof of Theorem 1.1 in Section 6.Actually, more precise estimates are obtained, cf.Theorems 5.1 and 6.1.
We also remark that as a part of our analysis the width of the inner layer near x = 0, where the singularity builds up, is estimated as O(e −µ(p−1)t ), and there the behavior is quasistationary in suitable rescaled variables.The precise statement of this stabilization result is given in Theorems 4.1 and 6.1.

Global existence
Henceforth, we will assume the conditions of Theorem 1.1.Under these assumptions it is clear from the Maximum Principle that a classical solution of problem (1.1) satisfies −C ≤ u(t, x) ≤ U (x) as long as it exists.
Our first result says that solutions are global in time when M = M c .

Proposition 2.1 Under the assumptions of Theorem 1.1, the solution is global in time
Proof.Assume for contradiction that T * < ∞.We know that gradient blow-up takes place only near x = 0.More precisely, since u(t, x) ≤ U (x) ≤ M = u(t, 1), we have u x (t, 1) ≥ 0 and we may apply [3, Lemmas 2.3 and 2.2] to deduce that (2.1) lim sup On the other hand, we have , we conclude from (2.2) and Hopf's Lemma that u x (t, 1) > U x (1) for all t ∈ (0, T * ].Fixing t 0 ∈ (0, T * ), one can then find M < M c close to M c and η > 0 small such that Moreover, due to (2.2) and u x (t 0 , 0) < ∞, we may also assume that

First estimates
The first step in the large-time analysis is a weighted L 2 convergence result.
Proposition 3.1 We have the following estimate: Remark 3.1.At this point, we make no claim about the sharpness of the exponent 2 above.Indeed, it will be improved in the end (as well as Proposition 3.2).
Proof.Let us put w = U − u.Then w ≥ 0. Using the inequality we see that w satisfies the inequality We note that, for each fixed t > 0, owing to (1.3), it holds We next obtain a first bound from below for the grow-up rate of u x .
4 Quasi-stationary analysis In this section, we perform the analysis of the behavior in a small region near x = 0 for large t.This is called in the technical literature the inner region analysis.We will closely follow the ideas of paper [11] to show that, when the solutions are properly re-normalized, the behavior is quasi-stationary.Let For each µ > 0, U µ (x) := µ 2−p U 1 (µ p−1 x), is the unique solution of Moreover, U µ (x) increases monotonically to U (x) as µ → ∞.Note that this is the same family V M but parametrized in terms of the slope at x = 0.
We shall use as rescaling parameter the quantity that diverges to ∞ as t → ∞, as we already know from Proposition 3.2.One can even prove that α(t) increases monotonically (see below).
We shall exhibit a quasi-stationary behavior, given by in a suitable boundary layer near x = 0.Moreover, we shall show that the stabilization is from above (this property will play a key role in the rigorous proof of the exact rate in Section 6).
More precisely, we have: Theorem 4.1 (i) For t large, α(t) is (strictly) increasing and in the sense that and uniformly for y = xα p−1 (t) ≥ 0 in bounded sets.
Proof of Theorem 4.1(i).For µ > 0, let J µ (t) be the number of intersections of u(t, •) with U µ in (0, 1) (i.e., the number of sign changes of U µ − u(t, •)).We shall use the known fact that J µ (t) is nonincreasing in time.
We note that so that the perturbation is not integrable in time.However, the following lemma ensures that the perturbation vanishes as τ → ∞.

Linearized operator and outer-region analysis
In connection with the linearization of equation (1.1) around the singular steady state U , we first need to study the singular eigenvalue problem (5.1) To this end, for a given real k > 0, we introduce the Hilbert space We have the following.
Proposition 5.1 Let k > 0 and define Then λ is well-defined, λ ∈ (0, ∞) and there exists ϕ ∈ H which solves the minimization problem (5.2) and enjoys the following properties.
With Proposition 5.1 at hand, we may formulate the main result of this section, which describes the asymptotic behavior of u away from x = 0. We show that the exponential convergence rate is given by the precise constant λ just defined.
As a preliminary to the proof of Theorem 5.1, we need to study the following regular eigenvalue problem, as an approximation to the singular problem (5.1): (5.5) for each ε ∈ (0, 1).We denote by λ ε > 0 the first eigenvalue of problem (5.5).We have the following Lemma.
Remark 5.1.The full spectral synthesis of the singular operator x −k (x k w x ) x is not used for the control of the second order perturbation in the equation for w, unlike in [11].(Moreover it is not clear if such an approach would still work for a gradient nonlinearity.)Here, taking advantage of the first derivatives involved in the nonlinearity, this is replaced by the use of the Cole-Hopf trick (5.10) on each set [ε, 1].

Matching. Rate of blow-up
By comparing the estimates of Sections 4 and 5, we shall now be able to identify the asymptotic behavior of u x (t, 0).The following result actually summarizes the information we have obtained on the asymptotic behavior of u x , in inner and outer layers as well.Finally, the behavior (6.2) is a rewriting of (4.3), while the outer layer statement is immediate from Theorem 5.1 since the equation is not degenerate and the solution stabilizes.Remark 6.1.As in [11], the rigorous proof of the grow-up rate does not use the convergence statement for θ(s, y) defined in (4.6) (i.e.part (ii) of Theorem 4.1), but the fact that the stabilization is from above (part (i)).Remark 6.2.Inserting the obtained estimates into the equations satisfied by θ in Section 4 and w in Section 5, we can obtain increasingly accurate estimates for the asymptotic development of u and u x .We refrain from such an interesting extension in this work by reasons of space since we have not found a short rigorous argument.