Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate apbq (where a and b are the concentrations of A and B, respectively, with 0

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A → B at rate a p b q (where a and b are the concentrations of A and B, respectively, with 0 < p, q < 1) and well-posedness for this problem has been lacking up to the present.
2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. 1

. Introduction and motivation
In this paper, we consider an initial-boundary value problem arising as a generic model for a one-step autocatalytic reaction. The initial-boundary value problem is of semi-linear parabolic type, and in dimensionless form is given by u(x, 0) = u 0 (x) ∀x ∈ R ( 1.2) and u(x, t) is uniformly bounded as |x| → ∞ for t ∈ [0, T]. (1.3) The nonlinearity f : R → R is given by 4) and the initial data u 0 : R → R (which will be from a sufficiently smooth class of bounded functions) is such that ( 1.5) The indices p, q > 0 represent the reaction order. The chemical background of the model is reviewed in detail in [1]. Particular reactions which have been established as autocatalytic include the iodate-arsenic reaction, the acidic nitrate-ferroin reaction and the hydroxylaminenitrate reaction. Autocatalytic rate laws also arise in enzyme reactions (such as glycolysis) and in the calcium deposition in bone formation. Details on the occurrence of autocatalytic steps in biochemical reactions may be found in Murray [2, chs 5-7]. In a number of the above applications, it is possible that both 0 < p, q < 1. When p, q ≥ 1, the nonlinearity f : R → R is Lipschitz continuous on every closed bounded interval. In this case, the initial-boundary value problem (1.1)-(1.4) has been studied extensively. In particular, classical Hadamard well-posedness has been established, along with considerable qualitative information regarding the structure of the solution to (1.1)-(1.3). Specific attention has been focused on the convergence to the equilibrium state u = 1 via the evolution of travelling wave structures in the solution to (1.1)-(1.5) when the initial data is non-trivial, as t → ∞, their propagation speed, shape and form [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The cases when 0 < p < 1 and/or 0 < q < 1 have received much less attention, primarily because the nonlinearity f : R → R lacks Lipschitz continuity in these cases owing to the behaviour at u = 0 and/or u = 1 and the classical comparison theorems and continuous dependence theorems fail to apply. However, the case when 0 < p < 1 and q = 1 has been considered in some detail in [17][18][19] in the context of global existence and uniqueness, although full Hadamard well-posedness was not established. The qualitative features concerning the solution to (1.1)-(1.5) with non-trivial initial data, in this case, do not exhibit travelling wave structure, but uniform convergence over x ∈ R, to the equilibrium state u = 1, as t → ∞. This represents a significant bifurcation in the structure of the solution to (1.1)-(1.5) for p ≥ 1 and 0 < p < 1, respectively. It is the purpose of this paper to address the initial-boundary value problem (1.1)-(1.5) when 0 < p < 1 and 0 < q < 1. We establish, via novel comparison and continuous dependence theorems, global well-posedness and, under mild restrictions, uniform global well-posedness, together with detailed qualitative features relating to the solution to (1.1)-(1.5). The approach to achieving these results is based on the recent generic theory developed in [20,21] and relies heavily on these results. Qualitatively, we find that the global solution to (1.1)-(1.5) does not lead to the development of travelling wave structures. In the physical context in which the model (1.1)-(1.5) arises, this anomalous behaviour arises through the interaction of Fick's law with a singular behaviour in the reaction rate (f (u) → ∞ as u → 0 + ). This has been discussed in detail in [22][23][24][25][26] and references therein, where it has been proposed that a relaxation of this behaviour requires a suitable relaxation term to be included in a modified Fick's law.
The paper is structured as follows. In §2, we introduce the notation used within the framework of the theoretical study of semi-linear parabolic partial differential equations, as found in [21,27], and establish some elementary results, which will be of use in later sections. In §3, we establish that, for any initial data in the set of interest, there exists a global minimal solution and a global maximal solution to the initial-boundary value problem, via results contained in [20,21]. In §4, we obtain a uniqueness result via adapting methods and results contained in [21,28]. In §5, we obtain a continuous dependence result on the initial data for solutions to the initial-boundary value problem. In §6, we bring together these results to establish a statement about well-posedness, and address qualitative features of the solution to (1.1)-(1.5).

Problem statement and preliminaries
Here, we formally introduce the problem which this paper addresses together with notation and definitions which will be used throughout the paper. To begin, it is convenient to introduce the following sets: with T > 0. We also introduce the set of initial data U 0 as the set of all functions u 0 : R → R such that u 0 is bounded, continuous with bounded and continuous derivative and bounded and piecewise continuous second derivative. Additionally, we introduce the subset U 0+ ⊂ U 0 as Throughout the paper, we consider classical solutions u :D T → R to the following semi-linear parabolic Cauchy problem: where u 0 ∈ U 0+ , u ∈ C(D T ) ∩ C 2,1 (D T ) and f : R → R is given by with p, q ∈ (0, 1). For the initial data u 0 ∈ U 0+ , we define We refer to this initial-boundary value problem as (S) throughout the rest of the paper. For convenience, we introduce γ = p/(p + q) and observe that and that f : (−∞, γ ] → R is non-decreasing and f : [γ , ∞) → R is non-increasing.
In what follows, we denote by H α the set of functions g : R → R which are Hölder continuous of degree α ∈ (0, 1] on every closed bounded interval. In addition, a function g : R → R is said to be upper Lipschitz continuous when g is continuous, and for any closed bounded interval E ⊂ R, there exists a constant k E > 0 such that for all x, y ∈ E, with y ≥ x, This set of functions is denoted by L u . It is straightforward to establish that f : R → R as given by (2.1) satisfies f ∈ H α with α = min{p, q} and Hölder constant k H = 1 on every closed bounded interval E ⊂ R. In addition, we have while it follows from the mean value theorem that We are now in a position to address well-posedness of the problem (S) on U 0+ . Here, we adopt the definition of Hadamard, given by Given that (P1) and (P2) are satisfied for (S), then given any u 0 ∈ U 0+ and > 0, there exists a δ > 0 (which may depend on u 0 , T and ) such that, for where v :D T → R and u :D T → R are the unique solutions to (S) corresponding, respectively, to v 0 , u 0 ∈ U 0+ . This must hold for each T > 0.
When the above three properties (P1)-(P3) are satisfied, then (S) is said to be globally well-posed on U 0+ . Moreover, when (P1)-(P3) are satisfied by (S) and the constant δ in (P3) depends only on u 0 and (i.e. being independent of T), then (S) is said to be uniformly globally well-posed on U 0+ .
In what follows, it is also convenient to label as (Ŝ) that semi-linear parabolic Cauchy problem obtained from (S) by exchanging f : R → R as given by (2.1), withf : R → R. A regular subsolution and a regular supersolution to (S) or (Ŝ) will be as defined in [20, definition 4.1]. We first address the question of existence for the problem (S).

Existence
We now establish an existence result for (S). We first introduce the function f η : R → R, for any η ∈ (0, γ ], such that where f : R → R is given by (2.1). It follows from (2.5) that f η ∈ L u , and We next establish that (S), with any u 0 ∈ U 0+ , is a priori bounded onD T for any T > 0. We have the following.
after which an application of the extended maximum principle in [20, theorem 3.4] establishes that v ≤ 0 onD T , and so Because f η ∈ L u , a direct application of the comparison theorem in [20, theorem 4.4] establishes that u ≤ū onD T , and so as required.
Before stating the main existence result, we refer to [21, remark 8.4] for the definitions of a constructed maximal solution and a constructed minimal solution to (S) onD T . We now have the following.

Uniqueness
It is first instructive to consider the problem (S) when u 0 : R → R is given by u 0 (x) = 0 for all x ∈ R (and so u 0 ∈ U 0+ ). It is then straightforward to observe that u 1 :D ∞ → R, given by is a global solution to (S) in this case. However, now consider u 2 :D ∞ → R, given by It is readily verified that u 2 :D ∞ → R is also a global solution to (S) in this case. It follows that, in this case, (S) exhibits non-uniqueness. However, in what follows, with u 0 ∈ U 0+ , we establish uniqueness for (S).
It is convenient at this stage to introduce the following sup norms for the continuous and bounded functions v :D T → R and w : R → R as follows: Before we can establish a uniqueness argument, we first require an improved lower bound for solutions to (S). We have the following.
We can now establish a uniqueness result for (S). The proof follows a similar approach to that of Aguirre & Escobedo [28], with theorem 4.1 and the existence of the constructed minimal solution in theorem 3.2 playing a crucial role.
for all (x, t) ∈D T and any T > 0, on noting, via [21, corollary 5.16], that u c ,ū c :D ∞ → R are uniformly continuous onD T , and so (ū c − u c )(·, t) B is continuous for t ∈ [0, T]. Moreover, the right-hand side of (4.21) is independent of x, from which we obtain which gives, after an integration, where T k is defined in theorem 4.1 for k ∈ (0, 1), with k chosen so that On substituting (4.24) into (4.20), we have and so (4.26) on noting that the right-hand side of (4.26) is integrable via (4.22) and [21, corollary 5.16] with the limit of the right-hand side implied at t = 0. Next, we define the function w : [0, T k ] → R to be We now integrate (4.29) from s = to s = t (with 0 < < t ≤ T k ) to obtain Next, we substitute the bound in (4.22) into (4.27), which gives for 0 < < t ≤ T k . Finally, upon substituting (4.31) into (4.30), we obtain Following from the definition ofū c T k and u c T k , theorem 4.1 and (4.34), we have, for k ∈ (0, 1) as in (4.25), where u c T k (·, 0),ū c T k (·, 0) ∈ U 0+ via theorem 3.2 and [21, lemmas 5.12 and 5.15], because f ∈ H α . Moreover, from theorem 3.2 and (4.35), it follows that Additionally, both u c T k andū c T k are bounded, twice continuously differentiable with respect to x and once with respect to t onD T−T k . Now, because u c T k satisfies it follows from (4.36) and the extended maximum principle in [20, theorem 3.4] that Now, observe that becauseū c T k and u c T k solve (S) with initial data u 0 = u c (·, T k ) ∈ U 0+ , then, via (4.39), where f η : R → R is defined as in (3.1), with η chosen as Recall that f η ∈ L u , and also, via (4.40) and (4.36), u c T k :D T−T k → R andū c T k :D T−T k → R are a regular supersolution and a regular subsolution to (Ŝ) withf = f η : R → R and initial data u 0 =ū c (·, T k ) = u c (·, T k ) ∈ U 0+ . It follows from a direct application of the comparison theorem in [20, theorem 4.4 It then follows from (4.37) and (4.41) that This holds for any T > 0, and so u c =ū c onD ∞ , as required.
It has now been established that problem (S), with u 0 ∈ U 0+ , has a unique global solution, and, therefore, that (P2) is satisfied. We next consider continuous dependence on initial data u 0 ∈ U 0+ .

Continuous dependence
Here, we obtain a continuous dependence result for (S) on the set of initial data U 0+ . Before we can proceed with an argument, we require a comparison theorem, which arises as a consequence of the uniqueness theorem established in §4.
We can now consider continuous dependence of solutions to (S) with respect to the initial data u 0 ∈ U 0+ . We have the following.

Theorem 5.2 (Continuous dependence).
Given > 0, T ∈ (0, ∞) and u 10 ∈ U 0+ , there exists δ > 0, such that, for any u 20 ∈ U 0+ which satisfies u 20 − u 10 B < δ, the corresponding unique solutions u 1 , u 2 : D T → R to (S) are such that Proof. Consider u 30 ∈ U 0+ , given by with δ > 0. It follows from theorems 3.2 and 4.2 that there exists u 3 :D T → R that uniquely solves (S) with initial data u 30 ∈ U 0+ . Now, for any u 20 ∈ U 0+ such that u 20 − u 10 B < 1 2 δ, then with i = 1, 2. It then follows from taking u 3 :D T → R as a regular supersolution and u i :D T → R (i = 1, 2) as a regular subsolution to (S) with initial data u 30 ∈ U 0+ in theorem 5.1 that for all (x, t) ∈D T . Therefore, because the right-hand side of (5.4) is independent of x, we have from which we obtain (noting that ( Now, take δ sufficiently small so that T δ = δ (1−p) /(1 − p) < T and it follows from (5.6) that Next, fix k ∈ (0, 1) such that p < k < 1, and it follows, via theorem 4.1, that there exists T k > 0 which is independent of δ, such that Now, take δ sufficiently small so that T δ < T k , and set T > T k . From (2.5), then (5.8) and (5.3) establish that, for i = 1, 2, Proof. It follows from theorem 3.2 that there exists a global solution to (S) for any initial data u 0 ∈ U 0+ , and, thus, (P1) is satisfied. Moreover, via theorem 4.2, this solution is unique, and, hence, (P2) is satisfied. Finally, theorem 4.2 exhibits that for any > 0, T > 0 and u 0 ∈ U 0+ , there exists δ > 0 (depending upon , u 0 and T) such that, for all u 0 ∈ U 0+ that satisfy u 0 − u 0 B < δ, the corresponding solutions u :D ∞ → R and u :D ∞ → R to (S) satisfy u − u A < onD T , and, therefore, (P3) is satisfied. We conclude that the problem (S) is globally well-posed on U 0+ .
As a consequence of this we have the following. Proof. Follows directly from theorems 6.4 and 6. 6.