BLOW-UP BEHAVIOR FOR A QUASILINEAR PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITION

In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary, or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, we prove the convergence of the solution to the unique self-similar solution as t approaching the blow-up time.

1. Introduction. In this paper, we study the following initial boundary value problem (P): u t = u 1+γ u xx , 0 < x < 1, t > 0, (1.1) u x (0, t) = −u q (0, t), u x (1, t) = 0, t > 0, (1.2) u(x, 0) = u 0 (x), 0 ≤ x ≤ 1, (1.3) where γ > 0 and q > 0 are given constants, and u 0 is a positive bounded smooth function defined on [0, 1] such that u 0 (0) = −u q 0 (0) and u 0 (1) = 0. The local existence and uniqueness of positive solution of (P) can be derived by the standard theory of parabolic equation. We say that a solution u blows up in finite time T , if lim sup t→T − {max x∈ [0,1] u(x, t)} = ∞. The study of blow-up has attracted much attentions for past years. The typical questions are concerned about blow-up criteria, blow-up locations, blow-up rates, blow-up profiles, and so on. We refer the reader to the survey papers of Levine [18] and Deng-Levine [7], and the book by Samarskii-Galaktionov-Kurdyumov-Mikhailov [19]. Problem (P) with γ = 0 was studied by Ferreira-de Pablo-Rossi [8] for both the bounded interval and semi-infinite interval cases. For blow-up on the boundary, we refer the reader to the survey papers by Chlebík-Fila [4] and Fila-Filo [10]. 2
It is well-known that quenching problem is related to blow-up problem. Indeed, by setting u =û m , γ = −1/m, ξ = γx, τ = γ 2 t, the problem (1.4)-(1.6) becomes (1.1)-(1.3) with q = 1 and spatial domain [0, l/γ]. Therefore, in this paper we shall only consider the case when q = 1. Another related problem to (P) is about the blow-up behavior of the solution of the Cauchy problem for the equation where σ ≥ 1 and p > 1. We refer the reader to [13,14,15] and the references cited therein. On the other hand, the Cauchy problem for the equation (1.1) in higher spatial dimension has been studied by Bertsch-Ughi [2] and Bertsch-Dal Passo-Ughi for nonnegative initial data. See also [20] for a more general equation.
In studying the blow-up behavior near the blow-up time, it is crucial to analyze the so-called (backward) self-similar solutions of (P). Let T < ∞ be the blow-up time and assume that x = 0 is a blow-up point. For q > 1, we introduce the following self-similar change of variables: where the similarity exponents are given by Note that α > 0 and β > 0, if q > 1. It follows that u satisfies (1. Note that s → ∞ and R(s) → ∞ as t ↑ T − . We expect that, as s → ∞, the solution of (1.8)-(1.10) is stabilized. In this paper, we shall call a global solution of the following problem as a self-similar profile of (P): Since we are interested in the behavior of u as t ↑ T − , we shall be concerned with the positive global solution of (1.11)-(1.12). In particular, we are looking for a monotone decreasing positive global solution of (1.11)-(1.12). This paper is organized as follows. In §2, we shall derive a blow-up criterion, prove the single point blow-up for monotone solutions when q > 1, and study the case when q ∈ (0, 1). Motivated by a recent work [8], we shall prove in §3 that for q > 1 the self-similar profile exists and is unique, by using a phase plane analysis approach. Finally, by using a method of Zelenyak [21] (see also [15]), in §4, we shall prove the convergence of v, as s → ∞, to the unique self-similar profile for q > 1.

Blow-up Criterion and Location.
In this section, we first prove that the solution of the problem (P) always blows up in finite time. (2.1) Proof. By assumption, there is a positive constant δ such that u 0 ≥ δ in [0, 1]. Then, by the maximum principle, u(x, t) ≥ δ for the corresponding solution u of (P). We introduce the following quantity By differentiating N (t) and using (1.1)-(1.2), we get Since q > 0, we get Thus N (t) should vanish at some finite time. Therefore, the solution u cannot be bounded for all t > 0. This implies that there exists a finite T > 0 such that (2.1) holds and the theorem is proved.
In the following, we shall always assume that the solution u of (P) blows up at time T < ∞. For simplicity, from now on we shall further assume that Using (2.3), it is easily seen by the strong maximum principle that u x < 0, u xx > 0 and u t > 0. We say that a point x = a is a blow-up point, if there is a sequence {(x n , t n )} such that x n → a, t n → T − , and u(x n , t n ) → ∞ as n → ∞. Note that x = 0 is always a blow-up point. Proof. Suppose, for contradiction, that there exists another blow-up point a ∈ (0, 1]. Then any point b ∈ [0, a] is also a blow-up point, since u x < 0 and u t > 0. Now we fix any number b ∈ (0, a). Following [11], we consider the function Then it is easy to compute that by using the properties of h and the fact that By choosing ε small enough and using u Letting t ↑ T − , we reach a contradiction. Thus the theorem follows.
for all x ∈ (0, 1]. Since u(0, t) → ∞ as t → T − and 0 < q < 1, we conclude that u(x, t) → ∞ as t → T − for any x ∈ [0, 1]. This means that we have the blow-up in the whole domain. Moreover, we can estimate the blow-up rate for the case q ∈ (0, 1) as follows.
Hence we obtain Then the estimate (2.6) follows by an integration of (2.8) from t to T and using (2.7).
3. Self-similar Profile for q > 1. In this section, we shall study the solution of the initial value problem (1.11)-(1.12): From the local existence and uniqueness theorem of ordinary differential equations, it follows that there is a unique positive local solution g of (3.1)-(3.2) for each given initial value g(0) > 0. For convenience, let [0, R) be the maximum existence interval of g. Note that g > 0 in [0, R) and 0 < R ≤ ∞.
Since g = αg −γ > 0 when g = 0, we see that any critical point of g must be a local minimum point. Hence there is at most one critical point of g. Moreover, if g has a critical point y 0 > 0, then g (y) > 0 for any y ∈ (y 0 , R) and g (y) > 0 for any y ∈ [y 0 , R).
For a given solution g, define and so Later on in §4, we shall need the following property.
Proof. Note that g must be monotone decreasing to zero, under the assumption of the lemma. Integrating (3.1) from 0 to y, we get Using integration by parts, we have

JONG-SHENQ GUO
The lemma follows. We shall need the asymptotic behavior as y → ∞ of any monotone decreasing positive global solution of (3.1)-(3.2) as follows.

Lemma 3.2. For any monotone decreasing positive global solution
Proof. By assumption, we see that g(y) → L as y → ∞ for some L ∈ [0, ∞). We claim that L = 0. If L ∈ (0, ∞), then there exists {y n } → ∞ such that g (y n ) → 0 as n → ∞. Dividing (3.1) by y and integrating the resulting equation from 1 to y n , we obtain ∫ yn Hence the left-hand side of (3.5) is uniformly bounded for all n. But, for K large enough, we have ∫ yn a contradiction. Hence L = 0. Next, we claim that g (y) → 0 as y → ∞. For this, we set We claim that g (0) + I = 0. Since g < 0, by (3.3), g (0) + I ≤ 0. Since g(y) → 0 as y → ∞, there exists a sequence {y n } such that y n → ∞ and g (y n ) → 0 as n → ∞. = lim This completes the proof.

BLOW-UP BEHAVIOR 7
Following [1,17,8], we introduce the following variables: for any solution g of (3.1). Then it is easily to check that (U, V ) satisfies the first order autonomous system (Q): Note that there are two finite critical points A := (0, 0) and B := (1, 0) for the system (Q). Since the linearization of (Q) around A gives the matrix which has eigenvalues {1, 2} and corresponding eigenvectors {(1, 0), (α, 1)}, we see that A is an unstable improper node. In particular, it follows from an easy phase plane analysis that every orbit near A in the second quadrant of (U, V )-plane leaves A horizontally (see, e.g., [6]). Notice that orbits corresponding to monotone decreasing positive solutions of (3. Note that the critical point D of (Q) becomes the critical point E := (−α/β, 0) of (R) in the (U, W )-plane. It is easy to see that the linearization of (R) around E gives the matrix which has eigenvalues λ 1 = β > 0, λ 2 = 0, and corresponding eigenvectors v 1 = (1, 0), v 2 = ((1+α/β)(α/β), β). Hence the horizontal line is tangent to the unstable manifold of E. Since the center manifold is tangent to the eigenspace spanned by v 2 and dW/dτ < 0 for (U, W ) ∈ S, where by a standard technique (see, e.g., [5]), there exists a unique orbit of the system (R) tending to E as τ → ∞. This shows that there exists a unique orbit, call it as Γ * , of the system (Q) tending to the critical point D as z → ∞. Note that Γ * lies in the strip S for all large z. Since dU/dz < 0 on {V > 0, U = −α/β}, dU/dz > 0 on {V > 0, U = 0}, and dV /dz > 0 in the second quadrant, the orbit Γ * must tend to A as z → −∞.
We thus have proved the following existence theorem. We continue to prove the uniqueness of such solution. Note first that any orbit tending to A has the behavior V = bU 2 + O(U 3 ) as U → 0 − for some positive constant b (which depending on each orbit). Let b * be the constant corresponding to the orbit Γ * . Therefore, by the phase plane analysis, for each b > b * the corresponding orbit shall reach the positive V -axis in finite time and continue to stay in the first quadrant. These orbits are those solutions of (3.1)-(3.2) with exactly one critical point.
On the other hand, for each b ∈ (0, b * ) the corresponding orbit shall reach the half-line L := {V > 0, U = −α/β} in finite time. We claim that these orbits are those solutions of (3.1)-(3.2) which tend to zero in finite time. Let V = V (U ) be an orbit from A such that (U, V )(z 0 ) = (−α/β, c) for some c > 0. Note that V > c and U < −α/β for z > z 0 . Hence It follows from (3.11) that U (z) → −∞ as z → z − 1 for some finite z 1 > z 0 . Set y 1 := e z1 . Suppose for contradiction that g(y) > 0 for all y ∈ [0, y 1 ]. Then g (y 1 ) is finite by (3.3). This implies that U (z − 1 ) is finite, a contradiction. Hence we have proved that g(y) → 0 + as y → y − 1 . Therefore, we are ready to prove the following uniqueness theorem. Proof. Since we have a unique orbit in (U, V )-plane connecting the critical points A and D, it remains to show the one-to-one correspondence of orbits with the positive solutions of (3.1)- (3.2). This is equivalent to show that different values of g(0) give different orbits in S leaving from A. Given a positive constant b (which corresponding to an orbit in S leaving from A). Since by using (3.2), we obtain the one-to-one correspondence between b and g(0). Hence the theorem follows. In the following, we shall denote g * to be the unique monotone decreasing positive global solution of (3.1)-(3.2) and let µ * := g * (0).

Asymptotic Behavior Near
Blow-up Time for q > 1. In this section, we shall study the asymptotic behavior of the solution u of (P) near the blow-up time T . This is equivalent to study the stabilization, as s → ∞, of the solution v of (1.8)-(1.10). More precisely, we shall prove the following main theorem of this section. To prove this theorem, we shall divide our discussions into a few subsections as follows.

Some a priori bounds.
In this subsection, we shall derive some a priori bounds for v. First, we derive the following blow-up rate estimate.
For the upper bound, we compare u with the function where g 2 is the solution of (3.1)-(3.2) with g 2 (0) very large so that g 2 is decreasing to zero at some finite R and u 0 has at most one intersection with U 2 (x, 0). Note that this is possible, since, by Lemma 3.1, g 2 (y) → −∞ as y → R − . Note also that U 2 is defined only in the set {(x, t) | 0 ≤ x < R(T − t) β , 0 ≤ t < T }. Similar argument as above gives that u(0, t) < g 2 (0)(T − t) −α for all t ∈ (0, T ). The lemma follows. As a consequence of (4.1), we obtain the following estimate Since u xx > 0, we have v yy > 0. Hence, using (1.9) and (4.2), we obtain Also, u x < 0 implies that v y < 0 and so v ≤ κ. Using (4.2) and (4.3), it is easy to see that there is a positive constant δ ∈ (0, 1) such that v(y, s) ≥ a/2 for 0 ≤ y ≤ δ, s > s 0 . Indeed, given (y, s) with y ∈ (δ, e β(s−s + 0 ) ), s > s 0 , we can find an l ∈ (s + 0 , s) such that y = δe β(s−l) . Since v yy > 0, it follows from (1.8) that v s + βyv y + αv > 0.
Hence (4.5) follows by an integration of the above inequality from τ = l to τ = s. From (4.4) and (4.5), we can derive that polynomial growth estimates in y for v s and v yy , by applying the interior parabolic estimates to (1.8). More precisely, we have the following. for some positive constant C. Note that α/β = 1/(q − 1). We consider the function Then we have To estimate v s for a given (ȳ,s) withȳ 1, as in [15], we make the following change of variables: V (y, s) := Kv(µy +ȳ, µ 2 K 1+γ s +s), |y| ≤ 1, −1 < s ≤ 0, where k ≥ 1 is chosen so that 2k ≤ȳ ≤ 4k. Then V satisfies the equation Note that, by the choices of K and µ, we have 0 < µK 1+γ (µy +ȳ) ≤ 4µK 1+γ k ≤ 4 for |y| ≤ 1, Also, by using (4.7) and (4.5), we have for some constants c 0 and C 0 which are independent of (ȳ,s). By applying the interior Schauder estimate, we see that V s (0, 0) is bounded by a constant which is independent of (ȳ,s). This gives the estimate (4.6) and the lemma is proved.
From (1.8) and combining all the above estimates, the polynomial growth estimate in y for v yy can also be derived.

Backward problem.
To derive the convergence result, we need to construct a Lyapunov function. In constructing a suitable Lyapunov function, we first study the following backward initial value problem for a given (y, v, ξ) with y > 0, v > 0, ξ ∈ R: (4.10) The local existence and uniqueness of the solution of (4.9)-(4.10) near y is trivial. We call this local backward solution as g(z; y, v, ξ) or simply g(z). As before, we define Then ρ > 1, ρ < 0, and We first prove that this backward solution always stays bounded in [0, y]. Otherwise, if g(z) → ∞ as z → z + 0 for some z 0 ∈ [0, y], then g (z) → −∞ as z → z + 0 . On the other hand, since g ≥ δ in (z 0 , y] for some constant δ > 0, ρ is uniformly bounded in [z 0 , y]. It then follows from (4.11) that g (z + 0 ) is finite, a contradiction. Hence g remains bounded.
Taking any sequence {s n } with s n → ∞ as n → ∞, by the standard arguments (e.g., [12]), we conclude that a subsequence of the sequence {v n (y, s) := v(y, s+s n )} converges to the unique monotone decreasing positive global solution g * (y) of (3.1)-(3.2) as n → ∞. Since this limit is independent of the choice of {s n }, we conclude that v(y, s) → g * (y) as s → ∞ uniformly for any compact subset of [0, ∞). This completes the proof of Theorem 4.1.