On a $L^\infty$ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a $L^\infty$ functional derivative estimate for the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to the Cauchy problem for scalar semi-linear parabolic partial differential equations with a continuous nonlinearity $f:\mathbb{R}\to\mathbb{R}$ and initial data $u_0:\mathbb{R}\to\mathbb{R}$, of the form, \[ \sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \ \ \ \forall t\in [0,T] . \] Here $\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}$ is a functional as defined in \textsection 1. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence $(f_n,0,u^{(n)})$, where for each $n\in\mathbb{N}$, $u^{(n)}:\mathbb{R}\times [0,T]\to\mathbb{R}$ is a solution to the Cauchy problem with zero initial data and nonlinearity $f_n:\mathbb{R}\to\mathbb{R}$, and for which $\sup_{x\in\mathbb{R}} |u_x^{(n)}(x,T)| \geq \alpha>0$, with \[ \lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left( \sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)}) \right) \right) = 0 . \]

In the remainder of the paper, we consider the sharpness of a derivative estimate for solutions u :D T → R to the semi-linear parabolic initial value problem (T > 0) given by, with nonlinearity f ∈ C(R) and initial data u 0 ∈ BPC 1 (R). We consider bounded solutions to the initial value problem (12)-(13) (henceforth referred to as [IVP]), which are classical, in the sense that Related to [IVP], we introduce the set I T ⊂ A T such that We observe (for any T > 0) that I T is non-empty (take (f, v, u) ∈ A T with each being the zero function). We also observe, via [4], that for any T > 0, when f = f p : R → R (for any 0 < p < 1) is given by and u 0 : R → R is given by u 0 (x) = 0 for all x ∈ R, it has been established in [4] that there exists non-trivial u = u p :D T → R such that (f p , 0, u p ) ∈ I T .
We will examine [IVP] with f = f p in detail, in §2 and §4. Now, we consider a Schauder-type derivative estimate for [IVP], which is a straightforward extension of those given in [2,Lemma 5.12] and [3, Lemma 3.9].
Proof. Since u :D T → R is a solution to [IVP] with f and u 0 , it follows by definition that u|D t is a solution to [IVP] with f and u 0 onD t then for any 0 < t ≤ T , and hence, (f, u 0 , u|D t ) ∈ I t . For convenience we drop the restriction notation from here onward, (with (f, u 0 , u) ∈ A T , then (f, u 0 , u) ∈ A t for each 0 < t ≤ T ). Now, let (f, u 0 , u) ∈ I T . Then, since f (u) ∈ B T A , it follows via [2, Theorem 4.9] that, Again, since f (u) ∈ B T A and u 0 ∈ BPC 1 (R), we observe, via [2, Lemma 5.9] that both terms on the right hand side of (16) have continuous partial derivatives with respect to x on D T , which are given by, and so, Therefore, It follows from Lemma 1.1 that Therefore, from (20) and (21), we have, Since the right hand side of (22) is independent of x ∈ R the result follows.
A classical Schauder-type derivative estimate can now be obtained as follows, Proof. This follows directly from Proposition 1.3 and Proposition 1.2.
For each (f, u 0 , u) ∈ I T , we now have, via Proposition 1.3 and Proposition 1.2, that Therefore, given (f, u 0 , u) ∈ I T , then (||u x (·, t)|| B − F t (f, u 0 , u)) is bounded uniformly above and below for t ∈ [0, T ]. Moreover, via (23), it follows that for any (f, u 0 , u) ∈ I T , Now, motivated by Proposition 1.3 and (24), we refer to the derivative estimate in Proposition 1.3 However, we observe that this definition is not immediately satisfactory due to the following exam- Then, trivially, (f * , u * 0 , u * ) ∈ I T , with, Thus, it follows from (28) and (29) that Finally, it follows from (30) and (24) that and hence, the derivative estimate in Proposition 1.3, according to the above definition, is sharp. To remove such trivial cases, we introduce the following improvement to the above definition; namely, we refer to the derivative estimate in Proposition 1.3 as non-trivially sharp onD T when there exists α > 0 such that We can now state the main result in this paper, as It then follows immediately from Theorem 1.1 that Corollary 1.5. The derivative estimate in Proposition 1.3 is non-trivially sharp onD T for any T > 0.
The paper is structured as follows. In §2, for 0 < p < 1 we introduce (f p , 0, u (p) ) ∈ I T with u (p) :D T → R specific nontrivial anti-symmetric self similar solutions to [IVP], which correspond to front solutions in [4]. In §3, we consider a formal limit as p → 0 of a boundary value problem for the ordinary differential equation associated with u (p) . In §4, we establish Theorem 1.1. Finally in §5 we discuss alternative approaches to establish Theorem 1.1, as well as similar problems for which this type of theorem may be established.

The Problem (P p )
For p ∈ (0, 1), consider [IVP] with nonlinearity f p : R → R given by (15) and initial data u 0 : R → R such that u 0 = 0. Henceforth we will refer to this as (P p ). In [4,Theorem 3.14] it is demonstrated that, for any T > 0, there exists a self similar solution u (p) :D T → R to (P p ) of the form with η = xt −1/2 whilst w p : R → R is such that w p ∈ C 2 (R), and The function w p : R → R, for p ∈ (0, 1), will be used extensively throughout the rest of the paper. Now, since u (p) :D T → R given by (31) is a solution to (P p ) for any T > 0, we have that In addition, it follows from (15), (31), (34) and (35), that, whilst from (31) and (36), Therefore, via (39), (40) and Proposition 1.3, where φ : (0, 1) → R is given by Furthermore, it follows that the inequality in (41) is strict, by substituting into (21) which follows from (31), (34) and (35), and proceeding with the proof of Proposition 1.3. We observe that, φ(p) > 0 ∀p ∈ (0, 1), In addition, it follows from Proposition 1.2, with (15), (31) and (34), that, Then, via (39) and (45), we have, Now, we conclude from the discussion following (41) that We also observe from (31) and (37) that A proof of Theorem 1.1 will now follow, up to minor detail, if we are able to construct a sequence {p n } n∈N , such that p n → 0 as n → ∞, and It is the construction of such as sequence which we now address. However, before proceeding, it is worth noting from (41) and (37), that at this stage, we have, We proceed by examining the solution w 0 : R + → R to a boundary value problem in which the ordinary differential equation is a formal limiting form of that in (32), as p → 0 + . We then show that there exists a sequence {p n } n∈N , such that p n → 0 as n → ∞ and w pn → w 0 and w ′ pn → w ′ 0 uniformly on [0, X] as n → ∞, for any X > 0. The result then follows on observing that w ′ 0 (0) = 2/ √ π.

The problem (S 0 )
In this section, we examine the problem given by taking the formal limit as p → 0 in the initial value problem for the ordinary differential equation studied in [4]. We seek a function w : [0, ∞) → R such that w ∈ C([0, ∞)) ∩ C 2 ((0, ∞)) and We refer to this linear inhomogeneous boundary value problem as (S 0 ). We observe that the coefficients in (51) are continuous functions of η ∈ [0, ∞). Thus, the homogeneous part (51) has two basis functions w 1 , w 2 : [0, ∞) → R and a particular integralw : [0, ∞) → R after which every solution of (51) may be written as, with A, B ∈ R being arbitrary constants. Inspection, followed by the method of reduction of order allows us to take for all η ∈ [0, ∞). It remains to apply conditions (52) and (53). These conditions are satisfied if and only if we choose Thus (S 0 ) has a unique solution w = w 0 : [0, ∞) → R given by where and we note that I(η) is monotone decreasing in η ∈ [0, ∞) with I(0) = √ π/8 (see [1, pp. 302, 7.4.11]), and I(η) decays exponentially as η → ∞. Finally, we observe from (54) and (55) that In the following section, we proceed to construct the sequence of functions w pn : R → R for which (50) holds.

Proof of Theorem 1.4
In this section, we construct a sequence {p n } n∈N such that p n ∈ (0, 1) for all n ∈ N, p n → 0 as n → ∞ and w pn : R → R satisfies for any X > 0, where w 0 : [0, ∞) → R is the unique solution to (S 0 ), given by (45). We note that via (57) and (56), we have which is crucial to the proof of Theorem 1.1. Throughout this section we consider w p : R → R restricted to the domain [0, ∞), so that w p = w p : [0, ∞) → R. To begin, we obtain uniform bounds on w p , w ′ p and w ′′ p for p ∈ (0, 1). We have first, Proposition 4.1. Consider w p : [0, ∞) → R with p ∈ (0, 1). Then, Proof. It follows from (34), (36) and (49) that for p ∈ (0, 1) Therefore, via the mean value theorem with (58), we have as required.
We immediately have, Corollary 4.13. There exists a subsequence {p n l } l∈N of {p n } n∈N such that Proof. It follows directly from (69) and Remark 4.6 that there exists a subsequence {p n l } l∈N of However, from Proposition 4.12, (54) and (56), and the proof is complete.

Discussion
We note here that it is not possible to establish a proof of Theorem 1.1 with a sequence of the form {(g n , u 0 , u (n) ) ∈ I T } n∈N with g n : R → R anti-symmetric, Lipschitz continuous, and such that g n (u) → 1 as n → ∞ for each u > 0, with u (n) :D T → R the unique solution to u (n) t − u (n) xx − g n (u (n) ) = 0 on D T u (n) = 0 on ∂D.
This follows since u (n) = 0 onD T for each n ∈ N, via uniqueness of solutions (see [2, Theorem 6.1]). However, we anticipate that a proof of Theorem 1.1 can be established, somewhat more generically, by considering a sequence of the form {(g n , u 0 , u (n) ) ∈ I T } n∈N , with g n and u (n) defined as above, but instead with non-zero initial data u 0 : R → R given by u 0 (x) = w 0 (x/λ 1/2 )λ ∀x ∈ R, for some fixed λ > 0, with w 0 given by (54)-(55). We anticipate that the approach adopted here can be used to show a similar sharpness result for the natural functional derivative estimate associated with solutions u : R N × [0, T ] → R to the Cauchy problem, Additionally, we note that it is likely that results of similar type to Theorem 1.1 can be established for functional derivative estimates of the Dirichlet and Neumann problems associated with (97)-(99).