Abstract
A no-flux initial-boundary value problem for the doubly degenrate parabolic system
is considered in a smoothly bounded convex domain \(\Omega \subset \mathbb {R}^n\), with \(n\ge 1\) and \(\ell \ge 0\). The first of the main results asserts that for nonnegative initial data \((u_0,v_0)\in (L^\infty (\Omega ))^2\) with \(u_0\not \equiv 0\), \(v_0\not \equiv 0\) and \(\sqrt{v_0}\in W^{1,2}(\Omega )\), there exists a global weak solution (u, v) which, inter alia, belongs to \(C^0(\overline{\Omega }\times (0,\infty )) \times C^{2,1}(\overline{\Omega }\times (0,\infty ))\) and satisfies \(\sup _{t>0} \Vert u(\cdot ,t)\Vert _{L^p(\Omega )}<\infty \) for all \(p\in [1,p_0)\) with \(p_0:=\frac{n}{(n-2)_+}\). It is next seen that for each of these solutions one can find \(u_\infty \in \bigcap _{p\in [1,p_0)} L^p(\Omega )\) such that, within an appropriate topological setting, \((u(\cdot ,t),v(\cdot ,t))\) approaches the equilibrium \((u_\infty ,0)\) in the large time limit. Finally, in the case \(n\le 5\) a result ensuring a certain stability property of any member in the uncountably large family of steady states \((u_0,0)\), with arbitrary and suitably regular \(u_0:\Omega \rightarrow [0,\infty )\), is derived. This provides some rigorous evidence for the appropriateness of (\(\star \)) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in (\(\star \)).
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1 Introduction
The question to which extent simple reaction-diffusion models can describe essential aspects of structure evolution in nature, and especially in living systems, has been stimulating considerable parts of the literature on parabolic problems over the past decades. Within the realm of bounded solutions, a dissipation-induced trend toward asymptotic equilibration has rigorously been confirmed to predominate in large classes of such systems ( [21, 12, 14, 22, 2] [9]). In line with this, substantial efforts have been focused on the identification of circumstances ensuring existence and favorable stability features of suitably structured steady states. Descriptions of structure-supporting properties on the basis of parabolic systems that exclusively involve reasonably regular ingredients seem accordingly constrained by natural limitations on richness of respectively relevant equilibria, despite a considerable collection of exceptions documented in the literature which assert the occurrence of impressively subtle stationary profiles especially near critical parameter settings (cf. [7, 4, 6, 8, 5, 24, 28] and also [25] for a very incomplete selection of findings in this regard).
The present study now addresses a modeling context which, despite a remarkable simpleness, seems characterized by an exceptionally large variety of evolutionary target states. Specifically, experimental findings reported in [10, 11] and [23] illustrate noticeably intricate facets in the collective behavior of populations of Bacillus subtilis when exposed to nutrient-poor environments; in particular, evolution into complex patterns, and even snowflake-like population distributions, appears as a generic feature rather than an exceptional event in such frameworks. Following experimentally gained indications for certain limitations of bacterial motility near regions of small nutrient concentrations, as a mathematical description for such processes the authors in [15] (cf. also [19] and [26]) propose the parabolic system
for the population density \(u=u(x,t)\) and the food resource distribution \(v=v(x,t)\). Forming the apparently most prominent ingredient herein, the diffusion operator in the first equation does not only degenerate in a standard porous medium-like manner at small values of u, but moreover accounts for said motility restrictions by containing a second degeneracy that emerges as soon as v approaches the level zero, in view of the second equation clearly expected to dominate at least on large time scales.
This latter peculiarity evidently brings about significant challenges beyond those mastered by classical approaches to degenerate diffusion problems of porous medium type ( [31, 2]), and also beyond those going along with the combination of two diffusion degeneracies due to the inclusion of diffusion rates that vanish both at small densities and at small gradient sizes of the diffusing quantity. In contrast to the situation in the latter class of doubly degenerate diffusion problems, meanwhile also fairly well-explored ( [13, 16, 29, 30]), especially the presence of a cross-degeneracy marks a considerable distinctiveness in (1.1). Accordingly, already at the stage of questions from basic existence theory the available knowledge so far seems limited to a statement on global solvability in a one-dimensional version ( [33]); built on a certain rather fragile energy structure, however, the corresponding arguments in [33] quite strongly rely not only on the presence of an additional cross-diffusive summand in the respective first equation and on strict positivity of \(\ell \), but beyond this also on the integrability assumption \(\int _\Omega \ln u(\cdot ,0)>-\infty \) which especially rules out an analysis of solutions emanating from compactly supported initial data.
Main results. The present manuscript now attempts to develop an approach which is not only capable of launching a fundamental theory of general nonnegative solutions for the simple system (1.1) in multi-dimensional settings, but which moreover also allows for conclusions addressing qualitative questions in such cases, in particular with regard to aspects related to stabilization of structures. For convenience setting \(D=d=\vartheta =1\) throughout the sequel, but keeping the parameter \(\ell \) as an arbitrary nonnegative parameter in order to include the proliferation-free case \(\ell =0\), we shall subsequently examine this in the framework of the initial-boundary value problem
in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\), with \(n\ge 1\), and with given initial data such that
It is fairly evident that in this context, any expedient analysis of (1.2) needs to appropriately quantify the potentially weakened diffusive action therein. In our first step toward this, by merely relying on basic \(L^1\) bounds for both components we shall use evolution properties of the sublinear functionals \(\int _\Omega u^q\) for \(q\in (0,1)\) to gain a priori estimates for expressions of the form \(\int _0^T \int _\Omega u^{q-1} v |\nabla u|^2\) within this range of q (Lemma 2.3), which will in turn imply bounds, inter alia, for
(Lemma 2.4). The core part of our analysis will then be concerned with the derivation of higher \(L^p\) bounds for u, where in the course of standard testing procedures a key requirement will consist in suitably exploiting the respective diffusion-induced contributions, as quantified through weighted expressions of type
This will be achieved in Lemma 4.3, on the basis of the bounds known for the integral in (1.4), by means of two functional inequalities of the form
and
valid with some \(C(K)>0\) for arbitrary positive \(\varphi \in C^1(\overline{\Omega })\) and \(\psi \in C^1(\overline{\Omega })\) fulfilling
throughout certain ranges of the parameters \(\alpha ,p,q\) and \(p_\star \) (Lemma 4.1 and Lemma 4.2).
Within the framework of an appropriately regularized variant of (1.2), the a priori information thereby generated will firstly enable us to make sure that for any fixed initial data fulfilling (1.3), the problem (1.2) is solvable by functions inter alia belonging to \(C^0(\overline{\Omega }\times (0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\) and enjoying some time-independent bounds in \(L^p(\Omega )\times L^\infty (\Omega )\):
Theorem 1.1
Let \(n\ge 1\) and \(\Omega \subset \mathbb {R}^n\) be a bounded convex domain with smooth boundary, and suppose that \(\ell \ge 0\) and that (1.3) holds. Then there exist functions
such that \(u\ge 0\) and \(v>0\) in \(\overline{\Omega }\times (0,\infty )\), and that (u, v) forms a global weak solution of (1.2) in the sense of Definition 2.1 below. Moreover, this solution has the additional boundedness property that
By suitably taking into account respective dependences on time, our estimates gained in the context of (1.4), (1.5), (1.6) and (1.7) will, besides implying the above, already pave the way for our subsequent asymptotic analysis. By means of a duality-based argument, namely, the latter can be seen to ensure a decay feature of the form
with \(C=C(u_0,v_0)\) and some \(\lambda >0\), and with \((W^{1,\infty }(\Omega ))^\star \) denoting the dual space of \(W^{1,\infty }(\Omega )\) (Lemma 5.1), and to thereby constitute the essential step toward our verification of the following statement on large time convergence of each individual among the trajectories obtained in Theorem 1.1.
Theorem 1.2
Let \(n\ge 1\) and \(\Omega \subset \mathbb {R}^n\) be a bounded convex domain with smooth boundary, let \(\ell \ge 0\), and suppose that (1.3) holds. Then there exists a nonnegative \(u_\infty \in \bigcap _{p\in [1,\frac{n}{(n-2)_+})} L^p(\Omega )\) such that as \(t\rightarrow \infty \), the solution (u, v) of (1.2) from Theorem 1.1 satisfies
and with some \(\gamma \in (0,1)\), as \(t\rightarrow \infty \) we have
Finally, the third of our main results provides some rigorous analytical evidence for the suitability of (1.1) in the considered modeling context: Namely, by adequately tracing the dependence of the constant \(C=C(u_0,v_0)\) in (1.10) on the initial data we can identify the following stability property enjoyed by actually any member of the uncountable family of function pairs \((u_0,0)\), which for all suitably regular \(u_0\) indeed form steady states of (1.2):
Theorem 1.3
Suppose that \(n\le 5\), and that \(\Omega \subset \mathbb {R}^n\) is a bounded convex domain with smooth boundary, and that \(\ell \ge 0\) and \(K>0\). Then for each \(\eta >0\), there exists \(\delta =\delta (\eta ,K)>0\) with the property that whenever (1.3) holds with
as well as
the solution (u, v) of (1.2) obtained in Theorem 1.1 satisfies
In particular, for the corresponding limit function \(u_\infty \) from Theorem 1.2 we then have
2 Approximation by positive classical solutions
The following solution concept to be pursued below seems to generalize notions of classical solvability in a manner fairly natural in contexts in which due to the considered diffusion degeneracy, no substantial first order information on the solution component u is expected.
Definition 2.1
Let \(n\ge 1\) and \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary, let \(\ell \ge 0\), and assume that \(u_0\in L^1(\Omega )\) and \(v_0\in L^1(\Omega )\) are nonnegative. Then a pair of nonnegative functions
such that
will be called a global weak solution of (1.2) if
for all \(\varphi \in C_0^\infty (\overline{\Omega }\times [0,\infty ))\) fulfilling \(\frac{\partial \varphi }{\partial \nu }=0\) on \(\partial \Omega \times (0,\infty )\), and if
for any \(\varphi \in C_0^\infty (\overline{\Omega }\times [0,\infty ))\).
In order to construct such solutions through some convenient approximation by classical solutions also in the presence of possibly discontinuous initial data, given \(u_0\) and \(v_0\) merely fulfilling (1.3) we first fix families \((u_{0\varepsilon })_{\varepsilon \in (0,1)} \subset W^{1,\infty }(\Omega )\) and \((v_{0\varepsilon })_{\varepsilon \in (0,1)} \subset W^{1,\infty }(\Omega )\) in such a way that
Then for each \(\varepsilon \in (0,1)\), the regularized variant of (1.2) given by
can be seen to actually inherit its initial non-degeneracy throughout evolution, and to thus allow for the following statement on global classical solvability by functions enjoying some rather expected basic features formally associated with (1.2):
Lemma 2.2
Let \(n\ge 1\) and \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary, let \(\ell \ge 0\), and assume (1.3) and (2.5). Then for each \(\varepsilon \in (0,1)\), there exist functions
such that \(u_\varepsilon >0\) and \(v_\varepsilon >0\) in \(\overline{\Omega }\times [0,\infty )\), and that \((u_\varepsilon ,v_\varepsilon )\) solves (2.6) in the classical sense. This solution satisfies
and
and
as well as
Proof
According to standard parabolic theory ( [1]), the regularity and positivity assumptions on \(u_{0\varepsilon }\) and \(v_{0\varepsilon }\) in (2.5) guarantee the existence of \(T_{max,\varepsilon }\in (0,\infty ]\) and positive functions \(u_\varepsilon \) and \(v_\varepsilon \) from \(C^0(\overline{\Omega }\times [0,T_{max,\varepsilon })) \cap C^{2,1}(\overline{\Omega }\times (0,T_{max,\varepsilon }))\) such that (2.6) is satisfied in \(\Omega \times (0,T_{max,\varepsilon })\), and that
By nonnegativity of \(u_\varepsilon v_\varepsilon \), the comparison principle ensures that this solution satisfies
and
where the latter especially asserts that
Another comparison argument therefore shows that writing \(c_3(\varepsilon ):=\Vert u_{0\varepsilon }\Vert _{L^\infty (\Omega )}\), we have
and that hence, in particular,
Again by comparison, we hence obtain that if we let \(y_\varepsilon \in C^1([0,\infty ))\) denote the solution of \(y_\varepsilon '(t)=-c_3(\varepsilon ) e^{c_2(\varepsilon )\ell t} y_\varepsilon (t)\), \(t>0\), with \(y_\varepsilon (0)=\min _{x\in \overline{\Omega }} v_{0\varepsilon }(x)>0\), then
Now assuming \(T_{max,\varepsilon }\) to be finite for some \(\varepsilon \in (0,1)\), we could combine (2.14) with (2.15) to infer that \(u_\varepsilon v_\varepsilon \) would then be bounded in \(\Omega \times (0,T_{max,\varepsilon })\), in view of standard parabolic regularity theory ( [17]) applied to the second equation in (2.6) implying that \(v_\varepsilon \) would belong to \(C^{1+\theta _1,\frac{1+\theta _1}{2}}(\overline{\Omega }\times [\frac{T_{max,\varepsilon }}{2},T_{max,\varepsilon }])\) for some \(\theta _1=\theta _1(\varepsilon )\in (0,1)\). Since \(u_\varepsilon v_\varepsilon \) would then also be uniformly bounded from below by a positive constant in \(\Omega \times (0,T_{max,\varepsilon })\) by (2.13) and (2.16), and since thus the first equation in (2.6) would actually be uniformly parabolic in \(\Omega \times (0,T_{max,\varepsilon })\), we could employ the same token once again to infer from the first equation in (2.6) that \(u_\varepsilon \in C^{1+\theta _2,\frac{1+\theta _2}{2}}(\overline{\Omega }\times [\frac{T_{max,\varepsilon }}{2},T_{max,\varepsilon }])\) for some \(\theta _2=\theta _2(\varepsilon )\in (0,1)\). As thus \(u_\varepsilon v_\varepsilon \) would in in fact lie in \(C^{\theta _3,\frac{\theta _3}{2}}(\overline{\Omega }\times [\frac{T_{max,\varepsilon }}{2},T_{max,\varepsilon }])\) with \(\nabla (u_\varepsilon v_\varepsilon ) \in C^{\theta _3,\frac{\theta _3}{2}}(\overline{\Omega }\times [\frac{T_{max,\varepsilon }}{2},T_{max,\varepsilon }];\mathbb {R}^n)\) for some \(\theta _3=\theta _3(\varepsilon )\in (0,1)\), again relying on uniform parabolicity of both equations in (2.6) we could then invoke parabolic Schauder theory ( [17]) to obtain \(\theta _4=\theta _4(\varepsilon )\in (0,1)\) such that \(\{u_\varepsilon ,v_\varepsilon \} \subset C^{2+\theta _4,1+\frac{\theta _4}{2}}(\overline{\Omega }\times [\frac{T_{max,\varepsilon }}{2},T_{max,\varepsilon }])\), and that hence, in particular,
Together with (2.13) and (2.16), in view of (2.12) this would contradict the hypothesis that \(T_{max,\varepsilon }\) be finite. In consequence, (2.9) and (2.10) immediately result from (2.15) and (2.14), while (2.8) and (2.11) directly follow upon integrating in (2.6). \(\square \)
Throughout the remaining part of this manuscript, unless otherwise stated we shall tacitly assume that \(\Omega \subset \mathbb {R}^n\) is a bounded convex domain with smooth boundary, and that \(\ell \ge 0\), and whenever \((u_0,v_0)\) and families \(((u_{0\varepsilon },v_{0\varepsilon }))_{\varepsilon \in (0,1)}\) fulfilling (1.3) and (2.5) have been fixed, by \(T_{max,\varepsilon }\) and \((u_\varepsilon ,v_\varepsilon )\) we shall exclusively mean the objects introduced in Lemma 2.2.
2.1 Fundamental first-order estimates. Basic decay properties of v
The following simple observation forms a fundament for the main part of our first step toward deducing further regularity properties of the solutions gained above, to be accomplished in Lemma 2.4 below on the basis of the estimate in (2.17). Apart from that, (2.17) will later on be utilized as a basic piece of information in our derivation of \(L^p\) bounds for \(u_\varepsilon \) (Lemma 4.3), and (2.18) as well as (2.19) will prove useful in the course of our large time analysis in Lemma 5.5 and Lemma 5.7.
Lemma 2.3
Let \(n\ge 1\), assume (1.3) and (2.5), and let \(q\in (0,1)\). Then
and
and furthermore we have
Proof
Let \(\varepsilon \in (0,1)\). Then using that \(u_\varepsilon \) is positive in \(\overline{\Omega }\times (0,\infty )\), we may test the first equation in (2.6) against \(u_\varepsilon ^{q-1}\) to see that
which firstly implies (2.19) as an immediate consequence of the restriction \(q\in (0,1)\). After an integration in time, we moreover infer from (2.20) that thanks to the Hölder inequality,
As \(\int _\Omega u_\varepsilon \le \int _\Omega u_{0\varepsilon } + \ell \int _\Omega v_{0\varepsilon }\) for all \(t>0\) by (2.8), this establishes both (2.17) and (2.18). \(\square \)
Now in the course of a standard testing procedure performed on the second equation in (2.6), an application of (2.17) to \(q:=\frac{1}{2}\) enables us to control the respective interaction-driven contribution by means of conveniently decaying quantities. In particular, the following conclusion can therefore be drawn in such a way that besides providing first order regularity features, due to its temporally global nature it furthermore already includes some information on temporal decay in the second solution component; this latter aspect will be of essential importance to our argument asserting \(L^p\) bounds for \(u_\varepsilon \) (see Lemma 4.3). The following lemma is the only place in this manuscript in which convexity of \(\Omega \) is explicitly made use of.
Lemma 2.4
Let \(n\ge 1\). Then for all \(K>0\) there exists \(C(K)>0\) such that whenever (1.3), (2.5) and (1.13) hold, we have
and
Proof
As \(v_\varepsilon \) is positive in \(\overline{\Omega }\times (0,\infty )\) and actually belongs to \(C^3(\overline{\Omega }\times (0,\infty ))\) according to standard parabolic regularity theory, on the basis of the second equation in (2.6) we compute
Here we may combine the outcome of a straightforward rearrangement ( [32, Lemma 3.2]) with a known functional inequality ( [32, Lemma 3.3]) to see for all \(\varphi \in C^3(\overline{\Omega })\) such that \(\varphi >0\) in \(\overline{\Omega }\) and \(\frac{\partial \varphi }{\partial \nu }=0\) on \(\partial \Omega \), writing \(c_1:=\frac{1}{(2+\sqrt{n})^2}\) and using that \(\frac{\partial |\nabla \varphi |^2}{\partial \nu } \le 0\) on \(\partial \Omega \) by convexity of \(\Omega \) ( [20]) we have
Therefore, (2.23) implies that
and to derive the claimed conclusion from this, we now assume that (1.13) holds with some \(K>0\), observing that then
as well as
due to (2.5). Therefore, an application of Lemma 2.3 to \(q:=\frac{1}{2}\) shows that
while (2.11) entails that
Since by twice employing Young’s inequality we can estimate
upon an integration in (2.24) we thus infer that due to (2.25), (2.26) and (2.27),
and conclude as intended. \(\square \)
3 Global existence
Besides on the boundedness properties documented in (2.9), (2.10) and (2.21), our construction of a solution to (1.2) through a limit procedure in Lemma 3.2 will rely on some further information on significantly enhanced regularity within finite time intervals bounded away from the initial instant. Due to its temporally local nature, the following statement in this regard, based on a combination of comparison arguments and standard parabolic theories, can apparently not be used in our large time analysis below, and hence does not trace dependencies of the obtained constants on the size of the initial data.
Lemma 3.1
Let \(n\ge 1\), and assume (1.3) and (2.5). Then for all \(\tau >0\) and \(T>\tau \), there exist \(\theta =\theta (\tau ,T;u_0,v_0)\in (0,1)\) and \(C(\tau ,T)=C(\tau ,T;u_0,v_0)>0\) such that
and
as well as
Proof
We fix \(\tau >0\) and \(T>\tau \) and recall (2.9) to find \(c_1(T)>0\) such that
and that thus
According to a comparison argument, this firstly implies that if we let \((e^{t\Delta })_{t\ge 0}\) denote the Neumann heat semigroup on \(\Omega \), then
where observing that \(v_{0\varepsilon } \rightarrow v_0\) in \(L^1(\Omega )\) as \(\varepsilon \searrow 0\) due to (2.5) and the dominated convergence theorem, thanks to well-known regularization features of \(e^{\frac{\tau }{4}\Delta }\) we see that
Since our overall assumption \(v_0\not \equiv 0\) ensures that \(c_2(\tau )\equiv c_2(\tau ;u_0,v_0):=e^{\frac{\tau }{4}\Delta } v_0\) is positive throughout \(\overline{\Omega }\) by the strong maximum principle, we can thus pick \(\varepsilon _1=\varepsilon _1(\tau ;u_0,v_0)\in (0,1)\) in such a way that
so that, in particular,
because by the comparison principle and (2.5), \(e^{\frac{\tau }{4}\Delta } v_{0\varepsilon } \ge e^{\frac{\tau }{4}\Delta } (\frac{\varepsilon }{2}) = \frac{\varepsilon }{2}\) in \(\Omega \) for all \(\varepsilon \in (0,1)\).
Now a second application of (3.4), again accompanied by a comparison argument, shows that for arbitrary \(t\in (\frac{\tau }{4},T)\),
so that combining (3.5) with (3.6) we find that for any such t,
Besides trivially implying (3.3), in conjunction with (2.9), (2.10) and (2.5) this shows that in the identity
the functions \(a_\varepsilon :=v_\varepsilon \) and \(b_\varepsilon :=u_\varepsilon v_\varepsilon \) satisfy
and
whence a standard result on Hölder regularity in porous medium type scalar parabolic equations ( [27]) becomes applicable so as to provide \(\theta _1=\theta _1(\tau ,T;u_0,v_0)\in (0,1)\) and \(c_5(\tau ,T)=c_5(\tau ,T;u_0,v_0)>0\) such that
As quite a similar reasoning yields an analogous Hölder bound for the solutions \(v_\varepsilon \) of \(v_{\varepsilon t} =\Delta v_\varepsilon - b_\varepsilon (x,t)\), this entails that with some \(\theta _2=\theta _2(\tau ,T;u_0,v_0)\in (0,1)\) and \(c_6(\tau ,T)=c_6(\tau ,T;u_0,v_0)>0\) we moreover have
We may therefore draw on classical parabolic Schauder theory ( [17]) to finally find \(\theta _3=\theta _3(\tau ,T;u_0,v_0)\in (0,1)\) and \(c_7(\tau ,T)=c_7(\tau ,T;u_0,v_0)>0\) such that
which precisely establishes (3.2), while (3.1) has already been contained in (3.7). \(\square \)
A weak solution with the additional properties announced in (1.8) can now be obtained by means of a straightforward extraction using the Arzelà-Ascoli theorem.
Lemma 3.2
Let \(n\ge 1\), and assume (1.3) and (2.5). Then there exist \((\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)\) and functions u and v fulfilling (1.8) such that \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \), that \(u\ge 0\) and \(v>0\) in \(\overline{\Omega }\times (0,\infty )\), that
as \(\varepsilon =\varepsilon _j\searrow 0\), and that (u, v) forms a global weak solution of (1.2) in the sense of Definition 2.1.
Proof
According to the equicontinuity properties asserted by Lemma 3.1, the Arzelà-Ascoli theorem provides \((\varepsilon _j)_{j\in \mathbb {N}} \subset (0,1)\) as well as functions \(u\in C^0(\overline{\Omega }\times (0,\infty ))\) and \(v\in C^{2,1}(\overline{\Omega }\times (0,\infty ))\) such that \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \), and that \(u_\varepsilon \rightarrow u\) in \(C^0_{loc}(\overline{\Omega }\times (0,\infty ))\) and \(v_\varepsilon \rightarrow v\) in \(C^{2,1}_{loc}(\overline{\Omega }\times (0,\infty ))\) as \(\varepsilon =\varepsilon _j\searrow 0\). Clearly, u is nonnegative, while strict positivity of v in \(\overline{\Omega }\times (0,\infty )\) follows from (3.3). Apart from that, the estimates in (2.9) and (2.10) together with the Vitali convergence theorem imply that \(u_\varepsilon \rightarrow u\) and \(v_\varepsilon \rightarrow v\) in \(L^p_{loc}(\overline{\Omega }\times [0,\infty ))\) for all \(p\ge 1\) as \(\varepsilon =\varepsilon _j\searrow 0\), and that u belongs to \(L^\infty _{loc}(\overline{\Omega }\times [0,\infty ))\) and v lies in \(L^\infty (\Omega \times (0,\infty ))\). As (2.21) along with (2.10) ensures boundedness of \((\nabla v_\varepsilon )_{\varepsilon \in (0,1)}\) in \(L^4(\Omega \times (0,\infty ))\), we readily obtain that also \(v\in L^4_{loc}([0,\infty );W^{1,4}(\Omega ))\), and that (3.10) holds as \(\varepsilon =\varepsilon _j\searrow 0\).
Now the regularity requirements in (2.1) and (2.2) clearly result from the properties in (1.8) just asserted, and a verification of (2.3) and (2.4) can be achieved on the basis of (2.6) in a straightforward manner, using that due to (3.8) and (3.10) we have
as \(\varepsilon =\varepsilon _j\searrow 0\). \(\square \)
4 Degeneracy control via functional inequalities. \(L^p\) bounds for u
4.1 Two functional inequalities
Up to this point, temporally global bounds which exclusively refer to \(u_\varepsilon \), without any presence of weight functions involving \(v_\varepsilon \) as an expectedly decaying quantity such as in (2.18), seem limited to the \(L^1\) boundedness feature in (2.8). A key step toward a more substantial description of the large time behavior in (1.2), and especially toward an exclusion of asymptotic Dirac-type mass accumulation, will now consist in an adequate control of the diffusion degeneracy in the first equation from (2.6), and particularly its part stemming from the presence of the factor \(v_\varepsilon \) therein. In the framework of standard \(L^p\) testing procedures, to be performed in Lemma 4.3, this specifically amounts to appropriately estimating integrals of the form
from below, and to thereby control the temporal growth of \(\int _\Omega u_\varepsilon ^p\), as potentially driven by the forcing term \(\ell u_\varepsilon v_\varepsilon \) in (2.6).
Of crucial importance to our approach in this direction will be the following observation on how far expressions of the form in (4.1), when added to integrals of the type appearing in (2.22), dominate gradients of certain products involving \(u_\varepsilon \) and \(v_\varepsilon \).
Lemma 4.1
Let \(n\ge 1, \alpha >\frac{1}{2}, p_\star \ge 1\) and \(K>0\). Then there exists \(C(\alpha ,p_\star ,K)>0\) with the property that whenever \(\varphi \in C^1(\overline{\Omega })\) and \(\psi \in C^1(\overline{\Omega })\) are positive in \(\Omega \) and such that
for any choice of \(q\in [0,2\alpha -1]\) the inequality
holds.
Proof
We first observe that for any such \(\varphi \) and \(\psi \), the pointwise estimate
holds throughout \(\overline{\Omega }\) with \(c_1\equiv c_1(\alpha ,p_\star ):=(2\alpha )^\frac{2p_\star }{p_\star +2\alpha -1}\). Since \(\frac{p_\star +2\alpha -1}{p_\star }>1\) thanks to our assumption that \(\alpha >\frac{1}{2}\), we may employ the Hölder inequality and rely on (4.2) to control the integral of the second summand on the right of (4.4) according to
Similarly,
where using that the restrictions \(q\ge 0\) and \(q\le 2\alpha -1\) warrant that \(0\le \frac{p_\star (2\alpha -q-1)}{2\alpha -1} \le p_\star \), we may invoke Young’s inequality along with (4.2) to see that
In conjunction with (4.6) and (4.5), this shows that (4.4) implies the inequality
from which (4.3) immediately follows if we let
for instance. \(\square \)
Combined with suitable Sobolev embedding properties, the latter entails the following class of interpolation inequalities appropriate for our purposes.
Lemma 4.2
Let \(n\ge 1, p_\star \ge 1\) and
Then for all \(K>0\) one can find \(C(p_\star ,K)>0\) such that for any \(q\in [0,\min \{p_\star ,\frac{2p_\star }{n}\}]\) and for each \(\varphi \in C^1(\overline{\Omega })\) and \(\psi \in C^1(\overline{\Omega })\) fulfilling \(\varphi >0\) and \(\psi >0\) in \(\overline{\Omega }\) as well as
we have
Proof
We first consider the case \(n\ge 2\), in which \(W^{1,\frac{2n}{n+2}}(\Omega )\) is continuously embedded into \(L^2(\Omega )\), so that we can pick \(c_1>0\) such that
We may then apply Lemma 4.1 to \(\alpha :=\frac{2p_\star +n}{2n}>\frac{1}{2}\) to infer the existence of \(c_2(p_\star ,K)>0\) such that whenever \(\varphi \) and \(\psi \) are positive functions from \(C^1(\overline{\Omega })\) which satisfy (4.8), for any choice of \(q\in [0,2\alpha -1]\) we have
Apart from that, we note that as a consequence of the Hölder inequality, any such pair \((\varphi ,\psi )\) satisfies
because of (4.8). Together with (4.11) and (4.10), this already establishes (4.9) with \(C(p_\star ,K):=c_1 \cdot \max \big \{ c_2(p_\star ,K) \, , \, K^\frac{2}{n}\big \}\) in this case, for the restriction \(q\in [0,\frac{2p_\star }{n}]\) clearly warrants that \(q\le 2\alpha -1\).
If \(n=1\), we proceed quite similarly, replacing (4.10) with the inequality
valid for some \(c_3>0\) and all \(\rho \in W^{1,1}(\Omega )\), substituting (4.12) by the observation that for all positive \(\varphi \in C^1(\overline{\Omega })\) and \(\psi \in C^1(\overline{\Omega })\) we have
and applying Lemma 4.1 to \(\alpha :=\frac{p_\star +1}{2}\). \(\square \)
4.2 Global \(L^p\) estimates for u. Proof of theorem 1.1
Having Lemma 4.2 at hand, we are now prepared for our derivation of the following statement on \(L^p\) boundedness of \(u_\varepsilon \), yet conditional in presupposing the existence of corresponding bounds in \(L^{p_\star }\) with some \(p_\star \ge 1\), by means of the announced testing-based argument.
Lemma 4.3
Let \(n\ge 1\), \(p_\star \ge 1\) and \(K>0\), and let p be as defined in Lemma 4.2. Then there exists \(C(p_\star ,K)>0\) such that if (1.3), (2.5) and (1.13) hold, and if
then
and
as well as
Proof
We fix any \(q\in (0,1)\) such that \(q\le \min \{p_\star ,\frac{2p_\star }{n}\}\), and combine Lemma 2.3 with Lemma 2.4 and (2.11) to see that thanks to the hypothesis (1.13) we can find \(c_1(K)>0\) fulfilling
On the basis of (4.13) and our selection of p and q, from Lemma 4.2 we thus infer the existence of \(c_2(p_\star ,K)>0\) such that
As a multiplication of the first equation in (2.6) by \(u_\varepsilon ^{p-1}\), followed by an integration by parts, shows that
and that hence
according to (2.5) and (4.18), the claim directly follows. \(\square \)
Upon repeated application of the latter, we obtain bounds in all the \(L^p\) spaces appearing in the claim from Theorem 1.1. Apart from that, the additional decay features expressed in (4.20) and (4.21) will play essential roles in our derivation of stabilization in the first solution component in the next section (see Lemma 5.1).
Corollary 4.4
Let \(n\ge 1\), \(p\in (1,\frac{n}{(n-2)_+})\) and \(K>0\). Then there exists \(C(p,K)>0\) such that if (1.3), (2.5) and (1.13) are satisfied, then
and
as well as
Proof
We recursively define \((p_k)_{k\ge 0}\) by letting
Assuming (1.3), (2.5) and (1.13) to hold, from Lemma 4.3 we then immediately obtain \((c_k(K))_{k\ge 1} \subset (0,\infty )\) such that for each \(k\ge 1\),
as well as
Since it can readily be verified that \(p_k\nearrow \frac{n}{(n-2)_+}\) as \(k\rightarrow \infty \), for arbitrary \(p\in (1,\frac{n}{(n-2)_+})\) the estimates in (4.19), (4.20) and (4.21) easily result from (4.22) and (4.23) upon an interpolation using the basic bounds provided by (2.8), (2.17) and (2.11). \(\square \)
In particular, the latter completes our reasoning with regard to solvability and global \(L^p\) regularity in (1.2):
Proof of Theorem 1.1
The part concerning existence and regularity has been completely covered by Lemma 3.2. The additional boundedness feature in (1.9) immediately follows from (4.19) when combined with (3.8). \(\square \)
Remark
For initial data enjoying regularity and positivity features beyond those in (1.3), the existence result from Theorem 1.1 can be supplemented by a corresponding uniqueness statement by a straightforward combination of the reasoning from Lemma 2.2 with the standard theory developed in [1]: Indeed, if beyond (1.3) it was required that both \(u_0\) and \(v_0\) belong to \(\bigcup _{q>n} W^{1,q}(\Omega )\) and satisfy \(u_0>0\) and \(v_0>0\) in \(\overline{\Omega }\), then (1.2) could actually be seen to admit a global classical solution which is unique in the class of functions fulfilling \(\{u,v\} \subset \bigcup _{q>n} C^0([0,\infty );W^{1,q}(\Omega )) \cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\). Under the present mild hypotheses in (1.3) on the initial data, however, we do not expect solutions to be uniquely determined by the requirements in Definition 2.1, nor by the additional regularity features obtained in Lemma 3.2 which are yet fairly poor near the initial instant; we cannot even rule out the possibility that different choices of approximate initial data in (2.5) may lead to different limits.
5 Large time behavior
5.1 Large time convergence of u in dual Sobolev spaces when \(n\le 5\)
Fortunately, in all physically relevant space dimensions the respective restrictions on p in Corollary 4.4 are mild enough so as to allow for the following conclusion on large time decay of \(u_{\varepsilon t}\) on the basis of (4.20), (4.21) and, again, the basic integrability property from (2.11).
Lemma 5.1
Let \(n\le 5\). Then there exists \(\lambda =\lambda (n)>0\) such that for all \(K>0\) one can find \(C(K)>0\) with the property that if (1.3), (2.5) and (1.13) hold, we have
Proof
Since \(n\le 5\), we have \(3-\frac{n}{(n-2)_+} < \frac{n}{(n-2)_+}\), so that it is possible to pick \(p=p(n) \in (1,2)\) fulfilling
where using a continuity argument we may rely on the latter inequality in choosing \(r=r(n)\in (0,1)\) suitably small such that
Noting that the restriction \(p<2\) warrants that then \(\frac{3-p-r}{1-r}>1\), given \(K>0\) we can draw on Corollary 4.4 to see that according to (5.3) and the first inequality in (5.2) we can fix \(c_1(K)>0\) and \(c_2(K)>0\) such that whenever (1.3), (2.5) and (1.13) holds,
and
Under these hypotheses, we now use the first equation in (2.6) to find that for all \(t>0\) and any \(\psi \in W^{1,\infty }(\Omega )\) such that \(\Vert \psi \Vert _{W^{1,\infty }(\Omega )}\equiv \max \big \{\Vert \psi \Vert _{L^\infty (\Omega )} \, , \, \Vert \nabla \psi \Vert _{L^\infty (\Omega )} \big \}\le 1\),
so that
Since for all \(T>0\) and \(\varepsilon \in (0,1)\) we have
due to the Hölder inequality, from (5.4), (5.5) and (5.6) we thus infer that thanks to (2.11),
As
by (1.13) and (2.5), this implies the claim with \(\lambda (n):=\frac{r}{2}\) and \(C(K):=c_1^\frac{1}{2}(K) c_2^\frac{1-r}{2}(K) + \ell \cdot (K+1)^\frac{1-r}{2} |\Omega |^\frac{1-r}{2}\). \(\square \)
As (5.1) involves \(L^1\) norms with respect to the time variable only, in order to avoid any discussion of possibly measure-valued parts appearing in the time derivatives \(u_t\) of the corresponding limits we prefer to formulate a conclusion of Lemma 5.1 for u which is sufficient for our subsequent reasoning in a version including temporal BV norms only, and hence exclusively involving the zero-order expression u itself:
Corollary 5.2
Let \(n\le 5\) and \(K>0\), let \(\lambda =\lambda (n)\) and C(K) be as in Lemma 5.1, and assume (1.3), (2.5) and (1.13). Then for any \((t_k)_{k\in \mathbb {N}} \subset [0,\infty )\) such that \(t_{k+1}\ge t_k\) for all \(k\in \mathbb {N}\), we have
where we have set \(u(\cdot ,0):=u_0\).
Proof
Fixing any such nondecreasing sequence \((t_k)_{k\in \mathbb {N}}\), under the present hypotheses we infer from Lemma 5.1 that for each \(N\in \mathbb {N}\),
because \((t_k,t_{k+1})\cap (t_l,t_{l+1})=\emptyset \) for \(k\in \mathbb {N}\) and \(l\in \mathbb {N}\) with \(k\ne l\). Since from (2.5) and Lemma 3.2 we know that with \((\varepsilon _j)_{j\in \mathbb {N}}\) as provided by the latter we have \(u_\varepsilon (\cdot ,t_k) \rightarrow u(\cdot ,t_k)\) in \(L^1(\Omega ) \hookrightarrow (W^{1,\infty }(\Omega ))^\star \) as \(\varepsilon =\varepsilon _j\searrow 0\) for each \(k\in \mathbb {N}\), we obtain (5.7) from (5.8) upon taking \(\varepsilon =\varepsilon _j\searrow 0\) and then \(N\rightarrow \infty \) there. \(\square \)
When considering individual trajectories, we may here yet ignore any of the information about dependencies on initial data enclosed in (5.1), and to thereby obtain the following as a particular consequence.
Corollary 5.3
Let \(n\le 5\), and suppose that (1.3) and (2.5) hold. Then there exists a nonnegative \(u_\infty \in (W^{1,\infty }(\Omega ))^\star \) such that
Proof
We only need to observe that whenever \((t_k)_{k\in \mathbb {N}} \subset (0,\infty )\) is nondecreasing and unbounded,
and that thus (5.7) particularly asserts that \((u(\cdot ,t_k))_{k\in \mathbb {N}}\) forms a Cauchy sequence in \((W^{1,\infty }(\Omega ))^\star \). \(\square \)
Beyond this, however, the particular quantitative form of the right-hand side in (5.1) enables us to also make sure that initially small \(v_0\) enforce solutions to remain near \(u_0\) in their first component.
Corollary 5.4
Let \(n\le 5\) and \(K>0\). Then given any \(\eta >0\), one can find \(\delta =\delta (K,\eta )>0\) such that whenever (1.3), (2.5) and (1.13) hold as well as
we have
Proof
As the number \(\lambda (n)\) provided by Corollary 5.2 is positive, the claim immediately results upon applying the latter to \((t_k)_{k\in \mathbb {N}}\) defined by \(t_1:=0\) and \(t_k:=t\) for \(k\ge 2\) and any fixed \(t>0\). \(\square \)
5.2 Stabilization of \(u^\gamma \) in dual Sobolev spaces for all \(n\ge 6\) and some \(\gamma \in (0,1)\)
In higher-dimensional cases in which the estimates provided by Corollary 4.4 seem insufficient for the above argument, with regard to large time asymptotics we rather resort to an analysis of certain sublinear powers of u. Indeed, for suitably small \(\gamma \) the expressions \(u_\varepsilon ^\gamma \) can be seen to enjoy a stabilization feature similar to that in Lemma 5.1, albeit apparently without information about comparably favorable effects induced by small \(v_0\) here:
Lemma 5.5
There exists \(\gamma \in (0,1)\) with the property that whenever \(n\ge 6\) and (1.3) as well as (2.5) hold, there exists \(C=C(u_0,v_0)>0\) such that
Proof
We fix any \(\gamma \in (0,\frac{1}{2})\) and observe that then for all \(t>0\) and each \(\psi \in W^{1,\infty }(\Omega )\), according to (2.6) we have
from which it follows that with some \(c_1>0\),
for all \(T>0\) and \(\varepsilon \in (0,1)\). Since here
by Young’s inequality, the claim readily results upon combining (2.17) with (2.18) and (2.11), because both \(\gamma \) and \(2\gamma \) belong to (0, 1). \(\square \)
After all, convergence of such powers \(u^\gamma \) for each fixed trajectory can be asserted within this topological setting:
Corollary 5.6
Let \(n\ge 6\) and \(\gamma >0\) be as in Lemma 5.5, and assume that (1.3) and (2.5) hold. Then there exists a nonnegative \(u_\infty \in (W^{1,\infty }(\Omega ))^\star \) such that
Proof
This can be concluded from Lemma 5.5 in much the same manner as Corollary 5.3 was derived from Lemma 5.1. \(\square \)
5.3 \(L^1\) decay of v
By now making use of the lower bound for sublinear \(L^q\) quasi-norms from Lemma 2.3, irrespective of the spatial dimension we can derive the following decay property of v from (2.11) and (2.21).
Lemma 5.7
Let \(n\ge 1\) and assume (1.3) as well as (2.5). Then
Proof
We fix any \(q\in (0,1)\) such that \(\frac{4q}{3}\le 1\), and from (2.8) we then obtain \(c_1=c_1(u_0,v_0)>0\) such that
We moreover employ a Poincaré inequality to pick \(c_2>0\) fulfilling
and abbreviating \(c_3:=\frac{1}{|\Omega |} \int _\Omega u_0^q>0\), given \(\eta >0\) we choose \(t_0=t_0(\eta ;u_0,v_0)>1\) large enough such that
and
here we note that such a selection is possible due to the fact that (2.11), Lemma 2.4 and (2.10) together with Lemma 3.2 and Fatou’s lemma guarantee that both \(\int _0^\infty \int _\Omega uv\) and \(\int _0^\infty \int _\Omega |\nabla v|^4\) are finite, and that hence, by the Hölder inequality and again due to (2.10) and Lemma 3.2,
For \(t>t_0\) and \(\varepsilon \in (0,1)\), we may then decompose
where by the Hölder inequality, (5.15) and (5.14),
Since furthermore
according to Lemma 2.3 and the evident nonpositivity of \(\frac{d}{dt} \int _\Omega v_\varepsilon \) throughout \((0,\infty )\), from (5.18) we altogether conclude that
With \((\varepsilon _j)_{j\in \mathbb {N}}\) taken from Lemma 3.2, we let \(\varepsilon =\varepsilon _j\searrow 0\) here to infer from (2.5) and the locally uniform convergence properties of \((u_{\varepsilon _j})_{j\in \mathbb {N}}\), \((v_{\varepsilon _j})_{j\in \mathbb {N}}\) and \((\nabla v_{\varepsilon _j})_{j\in \mathbb {N}}\) asserted by Lemma 3.2 that in line with our definition of \(c_3\),
As \(\eta >0\) was arbitrary, this establishes the claim. \(\square \)
5.4 Proofs of theorems 1.2 and 1.3
Large time convergence in the claimed topological settings can now be obtained from Corollary 5.3, Corollary 5.6 and Lemma 5.7 by suitable interpolation using the boundedness features asserted by Theorem 1.1.
Proof of Theorem 1.2
As \(\sup _{t>0} \Vert v(\cdot ,t)\Vert _{L^\infty (\Omega )}\) is finite according to Theorem 1.1, for \(n\ge 4\) the convergence statement in (1.11) can be derived from Lemma 5.7 by a simple interpolation based on the Hölder inequality. In the case \(n\le 3\), we once again use that thanks to (2.21) and Lemma 3.2, the integral \(\int _0^\infty \int _\Omega |\nabla v|^4\) is finite, so that with some \(c_1>0\) and some \((t_k)_{k\in \mathbb {N}} \subset (0,\infty )\) fulfilling \(t_k\rightarrow \infty \) as \(k\rightarrow \infty \) we have \(\Vert v(\cdot ,t_k)\Vert _{W^{1,4}(\Omega )} \le c_1\) for all \(k\in \mathbb {N}\). Since for any such n the Gagliardo-Nirenberg inequality provides \(c_2>0\) fulfilling
in view of Lemma 5.7 this implies that
and that thus (1.11) also holds in this case, because being a classical solution of its respective sub-problem of (1.2) in \(\overline{\Omega }\times [1,\infty )\) by Theorem 1.1, thanks to the maximum principle the function v has the property that \(0<t\mapsto \Vert v(\cdot ,t)\Vert _{L^\infty (\Omega )}\) is nonincreasing.
To deduce (1.12), we only need to combine the outcomes of Corollary 5.3 and Corollary 5.6 with the observation that for each \(\kappa >0\) and any \(p\in (1,\frac{n}{(n-2)_+})\), the family \((u^\kappa (\cdot ,t))_{t>0}\) is bounded in \(L^\frac{p}{\kappa }(\Omega )\) by (1.9), and hence relatively compact with respect to the weak topology in this space. \(\square \)
Our main result on stability of arbitrary equilibria in (1.2) has actually been covered by Corollary 5.4 in its essence:
Proof of Theorem 1.3
It is sufficient to apply Corollary 5.4, and to note that with \((\varepsilon _j)_{j\in \mathbb {N}}\) taken from Lemma 3.2 we have
due to the evident nonincrease of \(0\le t\mapsto \int _\Omega v_\varepsilon (\cdot ,t)\) for all \(\varepsilon \in (0,1)\). \(\square \)
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The author warmly thanks the anonymous reviewer and Frederic Heihoff for numerous insightful comments which significantly improved this manuscript.
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Open Access funding enabled and organized by Projekt DEAL. He furthermore acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (Project No. 411007140, GZ: WI 3707/5-1).
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Winkler, M. Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calc. Var. 61, 108 (2022). https://doi.org/10.1007/s00526-021-02168-2
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DOI: https://doi.org/10.1007/s00526-021-02168-2