Selfsimilar solutions in a sector for a quasilinear parabolic equation

We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally"self-similar"in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.

Our motivation for studying this problem arises in flame propagation, motion by mean curvature and other applications (cf. [2,3,4,7,8,9,10,11] etc.). [2,3,9,10,11] etc. considered the linear boundary value problem: (1.2)    u t = a(u x )u xx , −ζ 1 (t) < x < ζ 2 (t), t > 0, u x (x, t) = −γ 1 , u(x, t) = −x tan β for x = −ζ 1 (t), t > 0, u x (x, t) = γ 2 , u(x, t) = x tan β for x = ζ 2 (t), t > 0, where γ 1 , γ 2 are constants. They gave details on the a priori estimates and on the existence of solutions of (1.2) with some initial data. Moreover, [2,3,11] proved that when γ 1 + γ 2 > 0, any time-global solution u is expanding (that is, it moves upward to infinity) and it converges to a selfsimilar solution: √ 2t ϕ x/ √ 2t . On the other hand, [9,10] proved that if γ 1 + γ 2 < 0 then any solution shrinks to 0 as t → T for some T > 0. Furthermore, if a is analytic then the rescaled solution u/ 2(T − t) converges to a backward/shrinking selfsimilar solution with the form: In this paper we will extend their results to spatially and temporally inhomogeneous boundary conditions: k 1 , k 2 ≡ const.. Our boundary conditions mean that the contact angles between the curve (the graph of the solution) and the boundaries of the sector domain depend on the spatial and temporal environment. Clearly, selfsimilar functions like √ 2t ϕ(x/ √ 2t) or 2(T − t) ψ x/ 2(T − t) is no longer a solution of (1.1). We have to adopt new concepts for the analogue of selfsimilar solutions. Our results in this paper show that problem (1.1) has a unique expanding solution which is selfsimilar at discrete times if k 1 , k 2 have special "similarity" (see (1.5)) and if min k 1 + min k 2 > 0. On the other hand, problem (1.1) has a shrinking solution which is selfsimilar at discrete times if k 1 , k 2 have special "similarity" (see (1.10)) and if max k 1 + max k 2 < 0.
It is easily seen that a necessary condition for the existence of a discrete expanding selfsimilar solution is that k 1 and k 2 are "similar" in a special way: (1.5) k i (t, u) = k i b 2 t, bu for t, u 0.
Next we consider selfsimilar solutions which shrink to 0 in finite time.
Our theorems extend the results about classical selfsimilar solutions in [2,3,9,10,11] to nonlinear boundary value problems. Our approach is essentially different from theirs though we will use their classical selfsimilar solutions as lower and upper solutions to give the growth bound for the solution of (1.1). We will convert the original problem by changing variables to some new unknowns in a flat cylinder, and then consider the ω-limits of the new unknowns. Using a result in [1] we show that for the problem of the new unknowns there are periodic solutions, which correspond to selfsimilar solutions of (1.1).
In Section 2 we give some preliminaries, including the selection of the initial data, comparison principles and the local existence result. In Section 3 we consider the expanding case and prove Theorem 1.1. In Section 4 we consider the shrinking case and prove Theorems 1.2 and 1.3.

Preliminaries
We use notation S := {(x, y) | y > |x| tan β, x ∈ R}, and use ∂ 1 S and ∂ 2 S to denote the left and right boundaries of S, respectively. Hereafter, for two functions u 1 (x), u 2 (x) with (x, u 1 (x)) ∈ S, when we write we indeed compare them on their common existence interval; when we write we mean that u 1 (x) u 2 (x) and the "equality" holds at some x.
For any t 0 > 0, let u(x, t) be a classical solution of (1.1) on time interval [0, t 0 ] with some initial datum. Denote
The following comparison principle follows from the maximum principle easily.
In the following sections we will use classical selfsimilar solutions of (1.2) as upper and lower solutions of (1.1) to give the growth bound for the solution u of (1.1)-(2.1).

Gradient bound of u
Proof. By (1.6), or (2.4) we have and Combining these inequalities with (2.3) we obtain u x tan β − σ by maximum principle. u x − tan β + σ is proved similarly.

Change of variables
To study the local and global existence of solutions of the initial boundary value problem (1.1)-(2.1), it is convenient to introduce new coordinates that convert the sector domain S into a flat cylinder. More precisely, we will make a change of The functions θ = θ(x, y, t), ρ = ρ(x, y, t) and s = s(t) are to be specified below. With these new coordinates, the function y = u(x, t) is expressed as ρ = ω(θ, s), where the new unknown ω(θ, s) is determined by the relation The function ω(θ, s) is well-defined provided that the map t → s(t) is strictly mono- We will see later that these monotonicity conditions always hold for the class of solutions which we consider. Indeed we will prove Once ω(θ, s) is defined, then substituting it into the relation y = u(x, t) yields . The expression (2.10) gives a formula for recovering the original solution u(x, t) from ω(θ, s). In order for u to be smoothly dependent on ω, we need the map θ → X(θ, ω(θ, s), s) to be one-to-one for each fixed s and that s → T (s) is strictly monotone for s ∈ [s 0 , s ∞ ). Indeed we will prove

Local existence
To get the local existence we first make the following change of variables. (2.12) The inverse map is Clearly, θ = θ 0 and θ = −θ 0 correspond to ∂ 2 S and ∂ 1 S, respectively.
Let u(x, t) > 0 be a classical solution of (1.1)-(2.1) for t 0, then defines a new unknown ρ = ω(θ, s) for s 0. This function is well-defined since Differentiating the expression e ω(θ,s) cos θ = u(e ω(θ,s) sin θ, t) twice by θ and once by t we obtain where ω 0 is defined by (2.14) at t = s = 0 and Thus cos θ + ω θ sin θ > 0 since it can not be zero by (2.17) and it is positive at θ = 0.
Considering the second inequality in (2.17) we have So Using the first inequality in this formula we have, for θ 0, Similarly, considering the first inequality in (2.17) we have and cos θ + v θ sin θ ε 1 for θ 0. Summarizing the above results we have By the standard theory for parabolic equations, we see that (2.15) has a classical solution on time interval s ∈ [0, 2τ ] for positive τ = τ (k 1 , k 2 , µ, ω 0 ). The second inequality in (2.19) implies that, once the solution ω of (2.15) is obtained then we can recover it to the original solution u of (1.1). In fact, Consequently, we have the following local existence result.

Expanding selfsimilar solutions
In this section we always assume that (1.7) holds.
Since the initial datum u 0 > 0, there exist t + , t − > 0 such that The comparison principle implies that

Changes of variables
In Subsection 2.5 we gave a local existence result. One difficulty for deriving the global existence is the lack of the growth bound for ω. To give the global existence we adopt another change of variables.
Let τ be the constant in Lemma 2.3 and let t − > 0 be as in (3.2). For any n ∈ N satisfying we introduce new variables by The inverse map is Clearly, θ = θ 0 and θ = −θ 0 correspond to ∂ 2 S and ∂ 1 S, respectively.

Bound of v
The local existence result Lemma 2.3 implies that v exists on s ∈ [− 1 2 log n, 1 2 log(2τ + 1 n )]. We have changed u(x, t) to a new unknown v(θ, s). Similarly, we define v ± by ϕ ± in the following way e s e v ± cos θ = 2 e 2s − 1

By (3.3)
we have e s e v + cos θ u(e s e v + sin θ, e 2s − 1 n ). Noting e s e v cos θ = u(e s e v sin θ, e 2s − 1 n ) we have On the other hand, by the definition of v + we have (3.11) A similar discussion as above shows that The definition of v depends on n, it is not easy to give a uniform (for n) bound for v on [− 1 2 log n, ∞), but the above results show that a uniform (for n) bound for v is possible on [ 1 2 log τ, ∞).

Global existence
Now we consider problem (3.8) with initial datum v(θ, − 1 2 log n) = v 0 (θ), which is defined by (3.7) at s = − 1 2 log n. Using the bound and gradient bound in the previous subsections and using the standard theory of parabolic equations (cf. [5,6,12,13]) we can get the following conclusions.
where C 0 depends on k 1 , k 2 , µ and u 0 but not on s and n.
Indeed, studying the relations between v and u more precisely, it is not difficult to see that C 1 in this lemma can be replaced by C 2 √ t 0 + C 3 for some C 2 , C 3 depending on k 1 , k 2 , µ, u 0 and τ .
Now we use a result in [1].

Lemma 3.4 Let u(·, t) be a bounded (in H
where d, f, g * i are C 2 functions, T -periodic in t. Then there exists a T -periodic solution p(x, t) of (3.19) such that lim t→∞ u(·, t) − p(·, t) H 2 = 0. This lemma implies that the limit P of V n i is a log b-time-periodic solution of (3.15).
We now recover P to a solution of (1.1), the corresponding change of variables should be the limiting version as n → ∞ of (3.5) and (3.6), or for s ∈ (−∞, ∞).

Uniqueness of selfsimilar solutions
In this subsection we assume k i (t, u) ≡ k i (u) (i = 1, 2) and to prove the uniqueness conclusion in Theorem 1.1. The uniqueness for general k i (t, u) is still open. We begin with choosing a convex initial datum u 0 , that is, a(u 0x )u 0xx ǫ for some ǫ > 0. Such a choice is possible. For example, draw a line ℓ 1 from 0, tan β). Denote the contacting point between ℓ 1 and ∂ 2 S by A ′ 2 , then A ′ 2 := (x ′ , x ′ tan β) with ℓ 2 must contact ℓ 1 at some point A 3 ∈ S provided ǫ ′ > 0 is small enough. Now we smoothen A 1 A 3 + A 3 A 2 such that the smoothened curve C is strictly convex, it is tangent to A 1 A 3 at A 1 , tangent to A 3 A 2 at A 2 . Now the corresponding function u 0 of C is a desired initial datum. Let u be the solution of (1.1) with above constructed initial datum u 0 . Denote where f 1 and f 2 are continuous functions. Maximum principle implies that η = u t > 0 for t > 0. Using the same notions as in previous subsections we have 1 + v s > 0 and so 1 + V s 0 for s 1 2 log τ . Using (3.17) one has 1 + P s 0 for all s ∈ R, this implies that U t 0. Finally, the strong maximum principle implies that U t > 0 for all t > 0.
Since both U and U * are selfsimilar we have Hence (3.20) implies that On the other hand, by comparison principle Lemma 2.1 and (3.20) we have Therefore, (U, Ξ 1 , Ξ 2 ) is the unique selfsimilar solution of (1.1). This completes the proof of Theorem 1.1.

Shrinking selfsimilar solutions
In this section we always assume that (1.14) holds.
It is easily seen that the function with ζ i (t) = q i 2(T − t) (i = 1, 2) is a classical shrinking/backward selfsimilar solution of (1.2).
Proof. We first prove T + > T − . The areas D ± (t) of the regions enclosed by the graph of 2(T ± − t) ψ ± (x/ 2(T ± − t)), ∂ 1 S and ∂ 2 S are given by A simple computation shows that a(p)dp.
Next we prove T − δT + . For i = 1, 2, denote Q + i (resp. Q − i ) the end points of the graph of √ 2T
By (4.2) the graph of u 0 contacts the line segment Q + 1 Q + 2 . Since |u 0x | tan β−σ, we see that the graph of u 0 is above the line segment A 1 A 2 .
If for some T 0 > 0, the graph of is the contacting point between the line passing A 2 (resp. A 1 ) with slope tan β − σ (resp. − tan β + σ) and the left boundary ∂ 1 S (resp. right boundary ∂ 2 S). Therefore, Q − i are above B i for i = 1, 2. Using the coordinates of Q + i = ((−1) i r i cos β, r i sin β), where one can easily calculate the coordinates of B 1 and B 2 : The fact that Q − 2 is above B 2 implies that This proves the lemma.

Shrinking time for solutions of (1.1)-(2.1)
In this subsection we consider the shrinking time for the solution u = u(x, t; u 0 ) of (1.1) with initial datum u 0 . We give two results. The first one is about the shrinking time of u = u(x, t; u 0 ) for given u 0 , the second one is about the existence of u 0 for given shrinking time T .
By (4.3) we have r cos θ = u(r sin θ, t) √ 2T + max ψ + . So 0 r(θ, t) √ 2T + max ψ + / sin β. By (2.6) and (2.7) we have Thus the standard a priori estimates (cf. [5,6,12,13]) show that the solution of (4.6) with initial datum r(θ, 0) will not develop singularity till min r(·, t) → 0 as t → T for some T > 0. Moreover, T ∈ (T − , T + ) by (4.3) and Lemma 4.2. Now we prove (4.5). If u(0, T + 0) = 0 but u(x, T + 0) > 0. Without loss of generality we assumex > 0, then there existsx ∈ (0,x) such that This contradicts Lemma 2.2 and so the first limit in (4.5) holds. The last two limits in (4.5) follow from the first one. This proves the lemma. Proof. Choose two initial datum u ± 0 such that they satisfy conditions (2.2) and (2.3). Moreover, we choose u + 0 large such that (4.2) holds for some T − > T . By Lemma 4.3, the solution u(x, t; u + 0 ) shrinks to 0 as t → T + for some T + > T − > T . On the other hand, we choose u − 0 small such that (4.2) holds for some T + < T . By Lemma 4.3 again, the solution u(x, t; u − 0 ) shrinks to 0 as t → T − for some T − < T + < T . Now we modify the initial datum from u − 0 to u + 0 little by little such that the modified initial datum still satisfies (2.2) and (2.3). Since the solution u(x, t; u 0 ) of (1.1)-(2.1) depends on the initial datum u 0 continuously, we finally have an initial datum u 0 such that u(x, t; u 0 ) shrinks to 0 at time T ∈ ( T − , T + ).
In the following of this section, we fix T > 0 and choose the initial datum as in Lemma 4.4.
Therefore, problem (1.1) is converted into the following problem (4.11)
On the other hand, in the same time interval t ∈ [0, δ 2 T ] we have Thus by we have Therefore we obtain the bound of w for s ∈ [− 1 2 log T, − 1 2 log T − 1 2 log(1 − δ 2 )]: Note that the lower and upper bounds do not depend on s. Take a another pair T + * , T − * > 0 such that . Replacing T ± by T ± * in the above discussion we see that (4.13) holds on this time interval. Repeat such processes infinite times we obtain the estimate (4.13) for w on [0, T ).

A priori estimate for w
The gradient bound of w is similar as that for ω in Section 2 and that for v in Section 3. Using the standard theory of parabolic equations (cf. [5,6,12,13]) we can get the following conclusions.
Lemma 4.5 Problem (4.11) with initial datum w(θ, − 1 2 log T ) (which is defined by (4.9) at s = − 1 2 log T ) has a unique, time-global solution w(θ, s) where C depends only on µ, k i , σ and β but not on t, T and u 0 .
The global existence of w is not new, it has been obtained from the existence of r on [0, T ) in Subsection 4.2. The estimate is important and will be used below.
Finally, we prove the similarity of ( U, Ξ 1 , Ξ 2 ). By the similarity of ( U, Ξ 1 , Ξ 2 ) in the previous subsection and by the definitions of W and Υ i it is easily seen that