GENERAL INITIAL DATA FOR A CLASS OF PARABOLIC EQUATIONS INCLUDING THE CURVE SHORTENING PROBLEM

. The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When m ∈ (1 , 2], it is shown that there exists a smooth solution for t > 0 when the initial data belongs to L p loc ,p > 1. When m > 2, the same results holds when the initial data belongs to W 1 ,p loc ,p ≥ m − 1. An example is displayed to show that a smooth solution may not exist when the initial data is merely in L p loc ,p > 1. Solvability of the weak solution is also studied.


(Communicated by Juncheng Wei)
Abstract. The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When m ∈ (1,2], it is shown that there exists a smooth solution for t > 0 when the initial data belongs to L p loc , p > 1. When m > 2, the same results holds when the initial data belongs to W 1,p loc , p ≥ m − 1. An example is displayed to show that a smooth solution may not exist when the initial data is merely in L p loc , p > 1. Solvability of the weak solution is also studied.

1.
Introduction. Consider the one dimensional non-uniformly parabolic equation where a is a smooth function defined on R satisfying, for all p ∈ R, a(0) = 0, |a(p)| ≤ C 0 , C 0 a constant, and a (p) > 0 .
While (3) is imposed to ensure parabolicity, (2) that requires a be bounded globally is the most distinctive feature in this paper. A model case is given by that is, a (p) = (1 + p 2 ) −m/2 in (1). The special cases m = 2 and 3 correspond respectively to the curve shortening flow Gage-Hamilton [17], Grayson[20] and the one-dimensional case of the mean curvature equation studied in Gerhardt [18] and Ecker [11]. In Chou-Kwong [7] it is proved that the Cauchy problem for (1) under (3) and the condition ∞ a (s) admits a smooth solution for every smooth initial data. A preceding work is Ecker-Huisken [12] where the long time existence of the evolution of an entire graph under the mean curvature flow is established for any smooth initial graph. This property is in sharp contrast with uniformly parabolic equations such as the heat equation where its Cauchy problem is well-posed only for initial data under certain exponential growth.
Here we study to what extent that global solvability remains valid when the regularity of the initial data is substantially reduced. Recall the celebrated theorem of Widder [24] asserting that to every non-negative solution of the heat equation u, there associates a unique Radon measure µ satisfying and, conversely, there exists a unique non-negative smooth solution of the heat equation to each Radon measure µ satisfying this growth condition so that (6) holds. This theorem was extended to second order uniformly parabolic linear equations in divergence form in arbitrary dimensions in Aronson [3]. The measure µ is called the initial trace for the solution. Initial traces for the porous medium equation are studied in Aronson-Caffarelli [4], Bénilan et al [5] and Dahlberg-Kenig [9]. For the closely related evolution p-Laplacian equation one may consult also Zhao [26] and Wu, et al [25]. As the curve shortening flow is the Euclidean invariant version of the heat equation, it is natural to ask whether there is an initial trace for this or the more general equation (1). Indeed, it is not hard to show that the initial trace really exists. Specifically, for every non-negative solution of (1) under (2) and (3), there exists a Radon measure µ so that (6) holds. Yet unlike the heat equation, there is no growth restriction on the measure. Furthermore, it can be shown that the initial trace is a continuous measure, that is, µ({x}) = 0 for every x. Thus, for instance, the Dirac delta, which is the initial trace of the heat kernel, cannot be the initial trace of any solution to (1) under (2). Can we solve (1) under (2) and (3) for any given initial continuous Radon measure? This basic question has prompted us to a systematic study on the solvability of (1) for very general initial data. According to the Lebesgue decomposition of measures, every continuous Radon measure µ on the real line can be written as u 0 L 1 +λ where u 0 is locally integrable with respect to the Lebsegue measure L 1 and λ is a continuous singular Radon measure. Disregarding the singular part, our first result in this direction is Theorem A. Consider (4) where m ∈ (1,2]. For every u 0 ∈ L p loc (R), p > 1, there exists a solution u in C ∞ (R × (0, ∞)) which converges locally to u 0 in L p -norm as t → 0 + .
Here local convergence means that u(·, t) converges to u 0 in L p -norm on every bounded interval as t → 0 + .
The curve shortening problem was mainly investigated in the smooth category, see [17] and [8] for details. Nevertheless, there are some works for more general initial data. For instance, in Angenent [1,2], a general parabolic flow for closed curves on a surface whose normal velocity is driven by some curvature function is studied for initial curves which are either locally Lipschitz continuous or possess integrable curvature. In the level set approach for the mean curvature flow Evans-Spruck [13,14,15] and Chen-Giga-Goto [6], viscosity solutions are found for initial data which are the boundaries of bounded open sets. Now, according to Theorem A, in case of entire graphs, global solvability of the curve shortening problem is extended to all local L p -data, p > 1. Whether it could be extended to L 1 -functions and even continuous measures remains an open problem.
In Theorem A the model case for 1 < m ≤ 2 is studied, what happens to the range m > 2? As m increases, the relative strength of diffusion weakens. In order to retain solvability, we find that it is necessary to fortify the regularity of the initial data from L p loc -functions to W 1,p loc -functions.
) which converges to u 0 uniformly on every bounded interval as t → 0 + .
Comparing with Theorem A, one can see that the regularity requirement on the initial data has jumped from L p , p > 1, to W 1,p , p ≥ m − 1 when m goes beyond 2. Is this necessary? That is, once m goes beyond 2, in order to have classical solutions, do we need to push up the regularity of the initial data from L p loc to W 1,p loc ? The answer is affirmative. We will construct an initial W 1,p -function with p ∈ (1, m/2), m > 2, for which (4) does not admit any smooth solution. And this fact has led us to consider weak solutions for less regular initial data.
A function u ∈ C((0, T ); L 1 loc (R)), 0 < T ≤ ∞, is called a weak solution of (1) where a is bounded and continuous on R if its weak spatial derivative u x is locally integrable in R×(0, T ) and, for every ζ ∈ C 1 R×(0, T ) , vanishing outside I ×(0, T ) where I is a finite interval, and 0 < s < t < T , for positive δ. For every u 0 ∈ W 1,p loc (R), p > 1, there exists a weak solution in R × (0, ∞) which converges to u 0 uniformly on every bounded interval as t → 0 + .
Our approach to these theorems is quite standard, namely, first we approximate the initial data by a sequence of smooth initial data and solve the corresponding problem to obtain a sequence of approximating smooth solutions. Then, by establishing various a priori estimates, we can pass to limit to obtain the desired solutions. Theorems A, B and C will be proved in the first three sections. In Section 4, we construct a weak solution of (4) in W 1,p (R), 1 < p < m/2, for m > 2, whose gradient blows up at the origin for all time. It illustrates the fact that instant regularizing effect does not hold when the initial datum is not sufficiently regular. In the appendix, we discuss the existence of initial trace for the solution of (1) when a is continuous and bounded.
2. Some uniform estimates. In this section we derive some a prior estimates to be used in the proofs of the theorems. In order to obtain smooth solutions, we need uniform bounds on the solution as well as its gradient.
Throughout this paper let I ⊂⊂ J be two bounded, open intervals and d the distance from I to the boundary of J. Also let ϕ be a smooth function in J which is positive in the interior of J and vanishes at its boundary.
Our first estimate is concerned with a uniform bound of the solution in I × [t 1 , T ] in terms of the L p -norm of the initial data in J. Although it is primarily concerned with equation (4), we start by examining the more general equation (1) under (2) and (3). According to these assumptions, we have the structural inequality, for all p ∈ R, where α = max{a(1) −1 , |a(−1)| −1 } and β = max{a(1), |a(−1)|}.
Proof. We will establish the following two estimates. First, for each q ≥ p and t 1 ∈ (0, T ], there is some constant C 1 such that Next, for each t 1 ∈ (0, T ], there is some constant C 2 such that Clearly (10) follows from combining (11) and (12).
Step 1. For ε ∈ (0, 1], let u ε = (u 2 + ε) 1/2 . u ε is positive, smooth and tends to |u| uniformly in compact sets as ε → 0 + . For p > 1, we have Performing integration by parts, The right hand side of (13) can be estimated by Using this estimate in (13), after dropping the second integral on the left hand side, we have Integrating this differential inequality from s to t yields where C 4 depends on C 3 and p. Now, integrate (13) and use (14) and (15) In view of (9) and (p − 1) where C 5 depends further on C 4 , T, p and α.
Step 2. Recall the Gagliardo-Nirenberg interpolation inequality [16]: For a smooth function w on the bounded interval K, there is some constant C such that where 1/q = (1 − θ)/p, p > 1, and θ ∈ (0, 1). We choose q = p + 1 so that qθ = 1 and raise this inequality to the q-th power to get Next we replace p by p/(p − 1) in this inequality, fix the interval K, I ⊂⊂ K ⊂⊂ J and let w = u p−1 ε to get

KAI-SENG CHOU AND YING-CHUEN KWONG
Integrating from 0 to t, By (15)(s = 0) and (16) where ϕ is chosen so that it is identically equal to 1 on K, we arrive at where C 7 depends on C 4 , C 5 and C 6 .
On the other hand, we integrate (15) from 0 to t to get where this time we choose ϕ to be supported on K and equals to 1 on I. This estimate is valid for all p > 1. In particular, applying it to 2p − 1 and combining with (17), we finally obtain for some constant C depending on C 0 , α, d, t 1 , T, p and u(·, 0) L p (J) . We are ready to prove (11). Clearly it suffices to show it for q = p + k(p − 1) for all k ≥ 2. WLOG assume I = (x 0 − r, x 0 + r) and J = (x 0 − r − d, x 0 + r + d).
Let I j = (x 0 − r − jd/k, x 0 + r + jd/k) so that I 0 = I and I k = J and let s j = jt 1 /k, j = 0, · · · , k so that s 0 = 0 and s k = t 1 . Applying (18) to to the pair I j−1 and I j taking s j to be the initial time, we see that can be controlled by Therefore, beginning from j = 0, we can iterate this process and obtain our desired conclusion in k many steps.
Step 3. We prove (12). In view of (13), WLOG we may assume p ≥ 2. Taking p = 2 in (16) (replace 0, t by t 1 /2, T respectively) yields for some constant C. Now let A(p) be the primitive function of a satisfying A(0) = 0. We have Choosing ϕ equals to 1 on I and |ϕ x | ≤ 2/d, we have Integrating this inequality in s from t 1 /2 to t and applying (19) to the RHS, we have Now (12) follows after observing that for |p| ≥ 1, Next, we derive a uniform interior gradient estimate for the solution. (2), (3) and the following condition for some δ ∈ (0, 1) and p 0 . For every smooth solution of where C depends on δ, d, u L ∞ (J×[0,T ]) and u x (·, 0) L ∞ (J) . If in addition for some positive ρ, then holds where the constant C depends on δ, ρ, d and u(·, 0) L ∞ (J×[0,T ]) .
Proof. We adapt the method in [23]. Consider the function This function vanishes along the lateral boundary. Let us assume its maximum is attained at some ( At (x 1 , t 1 ), we have (log g) x = 0, (log g) xx ≤ 0, and (log g) t ≥ 0, that is, and By differentiating (1), we have Using (25), (27) and (26), By (1), this inequality simplifies to Moving the terms containing u 2 xx to the right hand side, the coefficient of u 2 xx is given by In view of (20), we can find some C 1 ≥ max{1, p 0 } depending on δ such that this coefficient is greater than δ 2 Now, we take η ≡ 1 to get Using (24) to eliminate u xx , For large u x , the last term in the RHS can be absorbed to the LHS. Hence, there is some constant Hence either δ 4 In both cases, we have g(x 1 , t 1 ) in controlled as asserted, and (21) follows. Next, we take η(t) ≡ t so that g vanishes also at t = 0. Now (28) becomes an inequality of the form A ≤ B + C where Then either A ≤ 2B or A ≤ 2C must hold. In the first case we control g(x 1 , t 1 ) as before. In the second case, by (24) and (22), Again this inequality is in the form A ≤ B + C. Both cases A ≤ 2B and A ≤ 2C lead to (23).
3. Existence of smooth solutions. In this section we prove Theorems A and B.  (2) and (3) such that, for every t ∈ (0, T ], u n (·, t) converges to some u(·, t) in local L p -norm for some p > 1. Suppose that {u n (·, 0)} converges locally to some u 0 in L p -norm. Then u(·, t) converges locally to u 0 in L p -norm as t → 0 + .

Proof.
Setting v = u m and w = u n for simplicity, we have, for ϕ as given in Section 1, By integrating this differential inequality and then letting ε → 0 + and m → ∞, we obtain Now, given ε > 0, there is some n 0 such that u n (·, 0) − u 0 L p (J) < ε for all n ≥ n 0 . In view of (29), there is some t 1 such that u n (·, t) − u(·, t) L p (I) < 2ε for all n ≥ n 0 and t ∈ (0, t 1 ]. By assumption, there is some t 2 such that u n0 (·, t) − u n0 (·, 0) L p (I) < ε for all t ∈ (0, t 2 ]. Therefore, for t, 0 < t ≤ min{t 1 , t 2 }, we have Proof of Theorem A. For u 0 ∈ L p loc (R), fix a sequence of smooth functions {u n0 } which converges to u 0 in L p loc -norm locally as n → ∞. Let u n be the corresponding smooth solution of (4) taking u n0 as its initial datum. The existence of this solution is guaranteed by theorem 2.2 in [7] (note that the a in [7] corresponds to a here). Using Proposition 1.1 and Proposition 1.2 (where (20) and (22) hold when m ∈ (1, 2]), {u nx } is uniformly bounded in every compact subset of R×(0, ∞). Observing that w = u nx satisfies the equation w t = (a (w)w x ) x , by standard parabolic theory, we obtain uniform estimates of all order on {u n } over every compact set in R × (0, ∞). By taking a diagonal subsequence, there is a subsequence {u nj } converging smoothly in every compact subset of R×(0, ∞) to a smooth function u in R×(0, ∞). Clearly, u solves (4). By Proposition 2.1, u(·, t) converges to u 0 in L p -norm on every bounded interval as t → 0 + , Theorem A holds.

Remark 1. According to proposition 2.3 in [7], the solution constructed in Theorem
A is unique when the initial datum belongs to L p loc (R) when p ≥ 2. It is also unique when u 0 ∈ L p (R), p > 1. To see this, let u 1 and u 2 be two solutions with the same initial function. First, by (1.3), they are uniformly bounded in L p -norm. Next, take v = u 2 and w = u 1 in the proof of Proposition 2.1 where we let ε → 0 + to get An integration yields We now choose ϕ such that ϕ(x) = 1, x ∈ (−R, R) and vanishes outside (−R − 1, R + 1) with |ϕ x | ≤ 2. As both u 1 and u 2 are in L p (R), the integral on the right hand side tends to 0 as R goes to infinity, so u 2 and u 1 are equal.

Remark 2.
By examining the proof of Theorem A, we see that this theorem holds also for the general equation (1) satisfying (2), (3), (20) and (22). Note that (5), which ensures the existence of smooth solutions for smooth initial data, holds under (20).
Next, we prove Theorem B.
Proof. We have By choosing a suitably small ε, we have for some ρ > 0. Now, take p = m − 1 and integrate from 0 to t to get Applying the mean value theorem to the second integral on the LHS in (0, t 1 /2), there is some τ ∈ (0, t 1 /2) such that At time τ , by Cauchy-Schwarz inequality, Now we choose ϕ equals to 1 on an interval K satisfying I ⊂⊂ K ⊂⊂ J. Using (31) and (32), we conclude where C depends on t 1 and u x (·, 0) L m−1 (J) . Finally, we apply Proposition 1.2 on K × [τ, T ] to deduce (30) from (21).
Proof of Theorem B With the help of Proposition 2.2, the proof of the existence of a smooth solution in R × (0, ∞) follows along the same lines as in the proof of Theorem A. Moreover, according to Proposition 2.1, the solution u(·, 0) converges to u 0 in L p on every bounded interval as t → 0 + . Using the fact that the local W 1,m−1 -norm of u is uniformly bounded (see (2.3)), by interpolation, Hence u converges to u 0 uniformly on every bounded interval. Uniqueness of the solution follows from proposition 2.3 in [7]. The proof of Theorem B is completed.

4.
Proof of Theorem C. In this section we prove Theorem C. Instead of the uniform estimates in Section 1, we need some integral estimates for this purpose. First, we have If in addition (8) holds, there is a constant C 2 further depending on δ (appearing in (8)) such that Proof. The first estimate can be obtained by letting ε → 0 + in (15). For the second estimate, letting w ε = (u 2 WLOG we may assume δ ∈ (0, 1) in (8). Using it, we have after choosing s = 2p/δ. Plugging this into (35), we have Ignoring the second integral on the left of (36) and integrating the inequality, we have and (34) follows by letting ε → 0 + in this inequality.
Proof. We multiply (1) by t 3/2 u t ϕ 2 and integrate to get,

KAI-SENG CHOU AND YING-CHUEN KWONG
On one hand, On the other hand, using A(p) ≤ pa(p), From (33), the L 2 -norm of u(·, t) can be controlled by the L 2 -norm of u(·, 0) locally. Therefore, the integral above is bounded by some constant multiple of t 1/2 . Putting things together, we have Choosing ϕ to be equal to 1 on K, an interval satisfying I ⊂⊂ K ⊂⊂ J, we have With (39) at our disposal, we compute where now ϕ is chosen to be equal to 1 on I and vanishes outside K. Integrating from s to t yields Multiply both sides of this inequality by s 3/2 and then use (39) to get Then a further integration over [0, t] gives (38).
Proof of Theorem C Let {u n } be a sequence consisting of smooth solutions to (1) in R × (0, ∞) whose smooth initial data {u n0 } converges to u 0 in local W 1,pnorm as n → ∞. The existence of these smooth solutions is due to theorem 2.2 in [7]. By (33), (34) and (38), for each bounded interval I and t 1 , T, 0 < t 1 ≤ T < ∞, {u n } is uniformly bounded in both C α (I × [t 1 , T ])(α = min{1/2, 1 − 1/p}) and L ∞ (0, T ; W 1,p (I)). Therefore, by Ascoli's theorem and Rellich-Kondrachov compactness theorem, by passing to a subsequence if necessary, we may assume there is some u belongs to both C α loc (R × (0, ∞)) and L ∞ loc (0, ∞; W 1,p loc (R)) such that (a) {u n } converges uniformly to u in every compact subset of R × (0, ∞) ; and (b) For each bounded I and T > 0, Note that (b) follows from (a) and (33). Each approximate solution u n satisfies the defining equation for the weak solution, namely, for ζ ∈ C 1 R × (0, T ) , vanishing outside I × (0, T ) where I is a finite interval, and 0 < s < t, In the lemma below, we will show that {u nx } converges to u x in L 1 (0, T ; L 1 loc (R)) for every T . It follows that we can find a subsequence converging pointwisely to u. Since a is bounded, applying Lebesgue's dominated convergence theorem and passing limit in (40), we conclude that u satisfies (7), that is, it is a weak solution to (1).
Finally, we claim that u(·, t) converges uniformly to u 0 on every bounded interval I as t → 0 + . Since u(·, t) is uniformly bounded in W 1,p (I) for t ∈ (0, 1], by interpolation it suffices to show its convergence in L 2 (I). We have Therefore, taking ϕ ≡ 1 on I, in view of (33) and (34), By letting n → ∞, we conclude as t → 0 + . The proof of Theorems C is completed except for the following lemma. Proof. Let v = u n and w = u m for simplicity. By differentiating the integral of (v − w) 2 ϕ over R and using (1), we obtain the estimate where C is independent of v and w. By (b), the RHS of this inequality tends to 0 as n, m → ∞. For M > 1 to be specified later, we express J × [0, T ] as the union of the following four subsets: First, by Similarly, observing that C(M + 1) > 0 on B, we have Next, since there exists a positive C(M ) so that for p, q, |q| ≤ M, |p| ≥ M + 1, or |p| ≤ M, |q| ≥ M + 1, we have Finally, by (34), These estimates together with (41) implies that where L is given by the RHS of (41). Letting m → ∞, we obtain where now We know that L 1 tends to 0 as n → ∞. Given ε > 0, we fix M so that C 1 /M p−1 < ε.
Then there exists some n 0 such that C 2 (L 1 + L 1/2 1 ) < ε. Hence, for all n ≥ n 0 , we conclude that

A genuine weak solution.
To show the necessity of studying weak solutions, we are going to construct a non-trivial weak solution of (4). In this section we take m > 2 and p ∈ (1, m/2), and show that for certain initial data, existence of classical solution is impossible. We note that interior blow-up of the gradient of solutions for equations similar to (4) is studied in Giga [18].
Step 1. The construction of sub-solution and super-solution. We set Substituting this into the equation we see that v is a solution to this equation if we choose With this choice of η, v is positive and concave on (0, ∞) with derivative blowing up at origin. It satisfies Consequently, v is a subsolution of (4) in (0, ∞) × (0, T ). We extend v(t) as an odd function to (−∞, 0]. Then it satisfies , and is therefore a supersolution on the negative side.
We will see that u is not a classical solution for t > 0. Indeed, assuming the contrary, suppose u is classical, then u(·, t) must be odd due to our uniqueness result in [7] (proposition 2.3).
Step 3. Finally, by the following proposition, since the derivative of v blows up at the origin, so does the derivative of u. This contradiction with our smoothness hypothesis on u shows that u cannot be a classical solution.
Step 2. Next we note that, The right hand side of (44) 2). This estimate remains valid when the variable T on the LHS is replaced by t. Integrating both sides with respect to t yields Invoking Hölder's Inequality, we obtain the key estimate Step 3. Now that for n ≥ 0, set We can express (45) in the form
Appendix . Initial trace. The following theorem asserts that the initial trace of a weak solution of (1) is the distributional derivative of a function in L ∞ loc (R), and in fact a continuous Radon measure when it is non-negative.
Theorem A.1. Consider (1) where a is bounded and continuous in R. For every weak solution u of (1), there exists an h in L ∞ loc (R) such that lim t→0 + u(x, t)ϕ(x)dx = − hϕ x dx, ∀ϕ ∈ C 1 c (R).
When u is non-negative almost everywhere, h is increasing, continuous and there is a continuous Radon measure µ such that − hϕ x dx = ϕdµ, ∀ϕ ∈ C 1 c (R).
Proof. Let [a, b] be any finite interval and δ > 0 a small number. We pick ζ = ζ(x) to be the test function which is equal to one on [a + δ, b − δ], vanishes at a and b, and linear on [a, a + δ] and [b − δ, b] respectively. Using ζ in (7) and then letting δ go to 0, we obtain b a u(x, t)dx − b a u(x, s)dx ≤ C|t − s|, 0 < s < t < 1, where the constant C depends only on a(p). Since a and b are arbitrary, the function U (x, t) ≡ x 0 u(y, t) dy is uniformly bounded in t ∈ (0, 1) on every finite interval. Moreover, the limit lim t→0 + U (x, t) exists at each x.
Let ϕ(x) = x a ψdx where ψ ∈ C c ((a, b)) is of zero mean. Then ϕ ∈ C 1 c ((a, b)) and we can plug it into (7) and perform integration by parts to obtain − U (x, t)ψ(x)dx + U (x, s)ψ(x)dx = t s a(U xx )ψ(x)dx.
Let h n be the function h corresponding to the interval (−n, n), n ≥ 1. From (47), it is clear that h n and h n+1 differ by a constant on (−n, n). By adding a suitable constant to each h n , n ≥ 2, we obtain an h ∈ L ∞ loc (R) for which (47) holds. Its uniqueness up to a constant is evident.
The function h is increasing when u is non-negative. To see this, denote the mollified ϕ by ϕ ε . By (49), we have