A strong maximum principle for nonlinear nonlocal diffusion equations

This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions. Solutions that attain interior global extrema must be identically trivial. However, depending on the nonlinearity, trivial solutions may not be constant in space; they may have an infinite number of discontinuities, for example. We give examples of nonconstant trivial solutions for different nonlinearities. For porous medium-type equations, these functions do not solve the associated classical differential equations, even those in weak form. We also show that these problems are globally wellposed for Lipschitz, nonnegative diffusion coefficients.

Our work concerns the following nonlocal nonlinear diffusion equation for function u : [0, T ] × R N → R: k u(t, x), u(t, y) u(t, y) − u(t, x) J(x − y) dy. (1.1) The kernel R n → R is an integrable, nonnegative function supported in the unit ball.The diffusion coefficient, or conductivity, k : R 2 → R is a locally Lipschitz continuous, nonnegative function.We seek solutions u(t, x) that are continuous in time but L ∞ in the spatial variables.Inside a bounded open set Ω ⊂ R N , u solves (1.1), but we set u(t, x) = ψ(x) on R N \ Ω.This is a Dirichlet condition in the nonlocal setting, analogous to setting u(t, x) = ψ(x) on ∂Ω in a classical (local) Dirichlet problem.Equation (1.1) is the nonlocal analogue of a classical nonlinear diffusion equation: where the diffusion coefficient k(u) accounts for nonlinear material properties, such as flow through a porous medium.Equation (1.2) can be obtained from (1.1) by rescaling the kernel J and taking the limit as a nonlocality "horizon" parameter vanishes, see [3,15].It can also be viewed as the first Taylor approximation to (1.1) [14], provided we take k(u, u) = k(u).
The integro-differential equation (1.1) contains, as a special case, the nonlocal linear diffusion equation where J(z) dz = 1 represents a probability density, and u(t, x) represents a density for some quantity, such as temperature [6] or population density [11].The first term represents the influx of species from a neighborhood surrounding the point x, while the second term accounts for population dispersion from x.
Equation (1.1) also includes other nonlocal diffusion equations, such as the nonlocal p−Laplace equation for p ≥ 2 covered in [3,Chapter 6]: |u(t, y) − u(t, x)| p−2 u(t, y) − u(t, x) J(x − y) dy, (1.4) and the fast diffusion equation of [3,Chapter 5]: Another physically interesting equation that can be obtained from (1.1) is analogous to the porous medium equation where m > 1; however, to our knowledge, this equation has not been studied in the literature.The closest that come to it are one using fractional differential operators [9], a nonlinear diffusion PDE with a nonlocal convection term [25], and one that is considered in [8]: Our work mostly concerns the strong maximum principle, a result that provides qualitative information about solutions as well as a tool for developing other a priori analytic estimates.For locally defined diffusion equations such as (1.2), the usual result for classical solutions If u attains a global extremum inside the parabolic cylinder [0, T ] × Ω, then u and its boundary values must be identically constant [18].
For integro-differential equations similar to the type in (1.1), there are positivity principles for the linear steady-state problem for L 2 functions [20] and strong maximum principles for linear and semilinear traveling wave equations [13,14], linear and Bellman-type-nonlinear parabolic problems [30], and degenerate parabolic and elliptic equations for semi-continuous functions involving Lévy-Itô-type nonlocal operators [12,24].
Our contributions are as follows: • Strong Maximum Principle: We formulate and prove a strong maximum principle for L ∞ solutions to equations of the type (1.1).Our results apply to, for example, solutions to a porous medium-type equa- Previous strong maximum principles for parabolic nonlocal diffusion equations have only dealt with continuous (or semi-continuous) solutions to equations with linear diffusion.
• Nonconstant Trivial Solutions: Our consideration of nonlinear diffusion shows that the maximum principle must be altered for certain types of nonlinearities.Instead of constant solutions, the maximum principle implies the presence of trivial solutions, which are, in general, not constant; they may have an infinite number of discontinuities.The precise nature depends on the nonlinearity k, and this is shown with examples.
• Wellposedness: We show that the Dirichlet problems considered are globally wellposed using a combination of a local result from the Banach fixed point theorem and the strong maximum principle.
In Section 2, we describe the initial boundary value problem and give a local wellposedness proof.Section 3 discusses nonconstant trivial solutions and gives examples for different nonlinearities.Section 4 gives the proof of the strong maximum principle.Finally, Section 5 includes the global wellposedness result.

Nonlocal nonlinear diffusion
After precisely outlining the initial-boundary value problem, we give a local wellposedness proof using Banach's fixed point theorem.

The initial-boundary value problem
Let Ω be a bounded open subset of R N and u : [0, T ] × R N → R be a function of time and space for some T > 0. We consider a nonlocal nonlinear diffusion equation: We solve (2.1) for u subject to a Dirichlet boundary condition and an initial condition The kernel J : R N → R is assumed to be nonnegative and integrable.We assume J(z) > 0 if |z| < 1, while J(z) = 0 for |z| ≥ 1.A common example in peridynamics is a radial kernel [15]: J(z) = J(|z|), but this is not necessary here.
The nonlinearity of the problem comes from k : R 2 → R. The conductivity k is assumed to be nonnegative.We take it to be locally Lipschitz continuous in the following sense: where K is nonnegative, nondecreasing, and continuous.The nonnegativity is imposed for global wellposedness.We will eventually specialize to classes of k defined by for p ≥ 3 (nonlocal p-Laplacian equation, see [3]), and k(u, v) = |u + v| m−1 for m ≥ 2 (nonlocal porous medium equation).We are most interested in the latter.We seek solutions u(t, x) to (2.1) that are in C([0, T ]) ∩ L ∞ (R N ).We assume that u 0 (x) and ψ(x) are in L ∞ (R N ) and L ∞ (R N \ Ω), respectively, and that u 0 (x) = ψ(x) when x is in R N \Ω.We are imposing continuity of u in time, which is the case in many physical applications.The requirement of u being in L ∞ as opposed to, say, L p , is because the strong maximum principle does not have a straightforward interpretation for unbounded functions.In addition, it allows us to consider more general, integrable kernels.
Using a standard Banach fixed point theorem technique, one can readily prove local wellposedness under the above assumptions.Global wellposedness is proven in a following section after more preliminary results are established.Some notations: Theorem 2.1 (Local wellposedness).We can find a sufficiently small T > 0 that depends on u 0 , J, and k such that there exists a unique B T solution u to the initial-boundary value problem (2.1)-(2.3).
Proof.We understand that, for each u 0 , a C[0, T ] solution u to (2.1) and (2.3) satisfies the following integral equation for t ≥ 0 and x in Ω: (2.5) For each w in B 0 , T ≥ 0, t ∈ [0, T ], and x in Ω, define A w,T : B T → B T as follows: A w,T u(t, x) := w(x) Our objective is to show that A w,T is a contraction mapping from B T,ǫ into itself for some T > 0 and some 0 < ǫ < ∞.If such parameters exist, then there exists a unique solution to the equation u = A u 0 ,T u (i.e.(2.5)) by Banach's fixed point theorem.
Remark 2.2.This result is not exhaustive by any means.It is possible to slightly extend this to those k that are (i) Lipschitz in the second argument, (ii) are differentiable away from u Still stronger results may also be obtained.See e.g.[3] for wellposedness results in the case of k(u, v) = |u − v| p−2 , where 1 ≤ p, reaction-diffusion equations [36], and distributional solutions to anomalous diffusion equations [16].Our analysis, while simple, was able to capture a large variety of nonlinearities k.For example, combined with subsequent analysis, we find that all nonlocal porous medium equations with conductivity k(u, v) = |u+v| m−1 , m > 2 are globally wellposed in the strong sense, which, for locally defined differential equations, requires much more effort; see e.g.[39].
We need the following fact to prove the Strong Maximum Principle.

Corollary 2.1 (Backwards wellposedness). There exists a unique solution
Proof.Setting u(t, x) = v(−t, x), we find that v satisfies (2.1) but with k → −k.Since this sign change does not affect the time T in (2.11), we conclude that v exists and is unique on in the above formulation, we obtain another Banach space, so the wellposedness result also holds for continuous functions.However, due to the nonlocal nature of (2.1), the continuity cannot, in general, be extended to the entirety of R N (see e.g.[20,30]).In other words, we set u = ψ on R N \Ω, rather than on R N \Ω.Note the space C(Ω) is the set of uniformly continuous functions on Ω extended to ∂Ω.Corollary 2.2.We can find a sufficiently small T > 0 that depends on u 0 , J, and k such that there exists a unique C([−T, T ], C(Ω), L ∞ (R N )) solution u to the initial-boundary value problem (2.1)-(2.3).
One interesting fact is that C([0, T ]) solutions to (2.1) that solve (2.5) automatically gain differentiability in time.
and we assumed k to be continuous.
Assuming that the first m time derivatives of u only have dependence on J, continuously differentiable functions of u, and integrals thereof, we differentiate (2.1) m times and find that the right hand side depends only on u, ∂ t u, ..., ∂ m−1 t u, and ∂ m t u, which, in turn, are in C[0, T ] ∩ L ∞ (Ω) by hypothesis.The result follows from induction up to m = n.
In particular, this is the case for the linear equation (1.3), and for (2.1) if k(u, v) = |u ± v| p , p = 2, 4, 6, 8, .... Remark 2.3.This type of result and proof is known for ODEs, such as u t = F (u), but does not follow so immediately in the classical setting for parabolic PDEs, such as u t = ∆u, since the right hand side, in that case, would also depend on the spatial derivatives of u (e.g.u tt = ∆ 2 u), which would not necessarily be time continuous.
We need the following related technical lemma for Theorem 4.1.
Proof.Choose t 1 and t 2 in [0, T ].Evaluating (2.1) at these times and subtracting gives: where we have put k i = k(u(t i , x), u(t i , y)), and u i z = u(t i , z), for brevity.Proceeding as in (2.7), we have the following estimate for the integrand: By (2.12), the L ∞ norm of u can be bounded in [0, T ], so this becomes: This means that u is a uniformly Lipschitz continuous function of time: Substitution into (2.13)gives the desired Lipschitz continuity:

Trivial solutions
When working with continuous solutions to differential equations, the usual notion of a trivial solution is the constant function.For example, the porous medium equation admits the trivial solution u ≡ constant.This is the solution obtained when applying the strong maximum principle to solutions with interior extrema.For nonlocal equations like (2.1) without spatial differential operators such as ∆ or ∇, this notion of triviality must be modified.There may be nonconstant trivial solutions.
Definition 1 (Trivial solutions).We say that u : R N → R is a trivial solution to (2.1) if it satisfies the following functional equation: where α is any function mapping R N into {−1, 1}, and U is a constant.Interestingly, α may be chosen so that u has infinitely many discontinuities.For example, if N = 1, the function α(x) = 1, x = 0, ±1/π, ±1/(2π), ... sgn(sin(1/x)), otherwise (3.4) has an infinite number of discontinuities in any neighborhood of x = 0.In general, functions defined by (3.3) are not even weak solutions to the classical porous medium equation (3.1).A weak solution u to (3.1) solves the following integral equation for every infinitely differentiable ϕ with compact support in (0, T ) ∩ R N .If, for example, we take N = 1, u to be that in (3.3), and α(x) = sgn(x), we find that the first term vanishes by the time-constancy of u, while, for the second term: which, clearly, is not always zero.For this choice of nonconstant α, (3.3) yields a weak solution only for U = 0.This example illustrates that nonlocal nonlinearities do not always give similar results to classical (local) ones.It also demonstrates that solutions to (2.1) do not always converge to those of (3.1) as, roughly, the support of J vanishes, which does hold in the linear (1.3) or p-Laplacian (1.4) cases [3].
This example may be contrasted with a less interesting one: which is equivalent, almost everywhere, to α(x) ≡ 1, and, hence, does give a weak solution to (3.1).
Remark 3.3.To our knowledge, nonconstant trivial solutions such as (3.3) have not been presented in the literature before.This is presumably due to the nonlinearities having been considered.For example, the p-Laplacian equation (1.4) considered in [3] only has one trivial solution: the constant function.It should also be noted that another porous medium-type equation, given by k(u, v) = |u| m−1 + |v| m−1 , admits only constant trivial solutions.
Remark 3.4.The general solution to (3.2) may be described as follows.For each u ∈ R, let F (u) ⊂ R be the set of those v such that k(u, v) = 0.If k(u(x), u(y)) u(y) − u(x) = 0 for any y = x, then either u(y) ∈ F (u(x)) or u(y) = u(x).Let U = u(z) for some z ∈ R N .Then, for each y = z, we have u(y) ∈ F (U) or u(y) = U.Thus, the image of u is {U} ∪ F (U).
It is clear that u(x) ≡ U is a solution, so assume that there exists some V = U such that u(x) = V on a set of positive measure.Then V ∈ F (U).We have k(U, V ) = 0, clearly, but in order for k(V, U) = 0, we need for U ∈ F (V ).Provided this is satisfied, a nonconstant solution to (3.2) exists.We may write it as follows: where α is any function from R N onto {0, 1}.Either u(x) = U, or u(x) = V .A necessary and sufficient condition for nonconstant solutions to exist is that U ∈ F (V ), provided that V ∈ F (U) \ {U}.A necessary condition for this to be true that U ∈ F (F (U)); a common sufficient condition is that Let us now consider a case for which the essential image of u has more than two elements.Suppose that u(x) = W on a set of positive measure, where W ∈ F (U) \ {U, V }.Clearly, we need for U ∈ F (W ), as with V .However, in order for k(V, W ) = k(W, V ) = 0, we need for V ∈ F (W ), and W ∈ F (V ).Since V, W ∈ F (U), we see that a necessary condition is that V, W ∈ F (F (U)).Therefore, the set F (F (U)) \ {U} must contain at least two elements for such solutions u to exist.
For example, if so a non-constant solution exists, namely, (3.3).In fact, since F (F (U))\{U} = ∅, there are no solutions u that attain three or more values, so (3.3) is the general solution.
A similar example to the previous is if k(u, v) = |a + uv| m for some a = 0, m > 0. In this and the previous case, F (U) defines an involutive function where defined.Indeed, F (U) = {−a/U} for each U = 0, so F (F (U)) = {U}.Again, nonconstant solutions exist.In this case, the nonconstant solution to can be presented as: where α is any function from R N into {0, 1}, and U = 0.A more complicated example is for k(u, v) = sin 2 π(u + v) 2 .We have This means that nonconstant solutions exist for every U ∈ R. Two-valued solutions to (3.2) are as follows: where α : R N → {−1, 1}, U is an arbitrary constant, and n is an arbitrary integer.
To see if there exists solutions u with images containing more than two elements, we consider W = −U + ℓ 1/2 ∈ F (U).For given U and V , the set has a solution n.In fact, this is the same equation that determines if the set F (W ) contains V .This equation has solutions only for countably many values of U.For instance, if U = π, then there are no solutions.As an example, suppose that U = 0, and that ℓ and m take values in the signed square integers, {p|p|} ∞ p=−∞ .Then this equation admits an integer solution n for every such ℓ, m.Therefore, we find the following solution to (3.2) where ν is an arbitrary function from R N into the integers (i.e ν(x) = ℓ 1/2 , ℓ = 0, ±1, ±4, ±9, ...).Remark 3.5.In the proof of the Strong Maximum Principle, the following functional equation appears: where U and χ ∈ Ω are constants.This is similar to (3.2), and, for the proof, we need to know under which conditions solutions to (3.7) necessarily satisfy (3.2).For a given y, either u(y) = U, or u(y) ∈ F (U) \ {U}.In order for k(u(x), u(y)) u(y) − u(x) = 0, we need for either u(y) = u(x), or u(y) ∈ F (u(x)).If u(x) = U, then we need for either u(y) = U or u(y) ∈ F (U).Both such conditions are satisfied as a result of u solving (3.7).On the other hand, if u(x) is in F (U) \ {U}, then we need for either u(y) = u(x), or for u(y) ∈ F (u(x)) ⊂ F (F (U)).If the first condition is not satisfied, then we need for u(y) ∈ F (u(x)) \ u(x), given that u(y) ∈ {U} ∪ F (U) \ u(x).
One important case is when F (U) = {u(x)}, and F acts like an involutive function: F (F (U)) = {U}, given that u(x) = U, since this implies that u(y) = U ∈ F (u(x)) = F (F (U)).As in Remark 3.4, examples for this case are k(u, v) = |u + v| m−1 (more generally, those that satisfy (k3) for the minus sign) as well as k(u, v) = |1 + uv| m .Thus, for the simple class of coefficients k in (k3), we have the following characterization.
Finally, trivial solutions satisfy the following obvious property that is important for proving the Strong Maximum Principle.Lemma 3.2.Let u be a solution to (2.1).If, for some t = t 0 , u(t 0 , x) is a trivial solution to (2.1) that solves (3.2), then u is identically trivial.
Proof.The function v(t, x) = u(t 0 , x) is a trivial solution to (2.1) for all times t.By forwards (Theorem 2.1) and backwards (Corollary 2.1) uniqueness of solutions, we can only conclude that u ≡ v is trivial.
We first consider continuous functions u ∈ C[0, T ] ∩C(Ω) ∩L ∞ (R N ).The proof is more technical in the more general L ∞ case, so this short proof is a useful illustration.Proof.Let (t 0 , x 0 ) ∈ (0, T ) × Ω be such that |u(t 0 , x 0 )| ≥ |u(t, x)| for almost all (t, x) ∈ [0, T ] × R N .By virtue of u(t 0 , x 0 ) being an extremum, we see that u t (t 0 , x 0 ) = 0. Substituting this and (t, x) = (t 0 , x 0 ) into (2.1)gives: Since u(t 0 , x 0 ) is a global maximum or minimum, the integrand is one-signed, so we deduce that it is identically zero.By Lemma 3.1, u(t 0 , y) is a trivial solution, which, from Lemma 3.2, implies that u is identically trivial.
We now consider the case of u being in C[0, T ] ∩ L ∞ (R N ).For future reference, we set: We recall the following well known result used in the proof of the Fréchet-Kolmogorov Theorem (see e.g.Theorem 2.21 in [1]).We also note another well known fact (Corollary 2.32 in [19]).
then there is a subsequence {f n j } such that f n j → f almost everywhere.Theorem 4.1 (Strong maximum principle).Let u be a C([0, T ], L ∞ (R N )) solution to (2.1).Suppose that k satisfies (k3).If either U + (t 0 ) ≥ u + (t) or U − (t 0 ) ≤ u − (t) for some t 0 ∈ (0, T ] and all t ∈ [0, t 0 ], then u is a trivial solution. Proof.Consider the first case of U + (t 0 ) ≥ u + (t).We have U + (t) = lim n→∞ u(t, x n (t)) for some sequence (x n (t)) ⊂ Ω.By the relative compactness of Ω, we may redefine (x n ) as one of its convergent subsequences: Let us evaluate (2.1) at t = t 0 , x = x n (t 0 ) and make a change of variables, y = z + x n (t 0 ), setting u n = u(t 0 , x n (t 0 )) for brevity: where the new integration domain over the unit ball comes from supp J = B 1 (0).
We can use this to deduce a positivity principle similar to those in [3,20].
Proof.If not, then, since ψ − ≥ 0 for all t, we must have U − (t 0 ) < 0 for some t 0 > 0. By continuity (Lemma 5.1), this means there exists a subinterval [a, b] ⊂ [0, t 0 ] on which U − (t) is decreasing and is negative.Of course, this means that U − (b) ≤ u − (t) for all t ∈ [a, b].By the strong maximum principle and solution uniqueness, this implies that u is a trivial solution for all time.However, since such solutions do not depend on time, this means that u = u 0 ≥ 0, a contradiction to u being negative.

Global wellposedness
Using the Strong Maximum Principle (Theorem 4.1), we first deduce that the L ∞ norm of u is a nonincreasing function of time.Combining this with the local wellposedness result (Theorem 2.1), we show that (2.1) is globally wellposed.
For the U + case, we need only note that U + (t) = −ess inf R N (−u(t, x)).It follows from the previous analysis that this is a continuous function.

Lemma 4 . 2 .
Let u : R N → R be an integrable function, and let τ a u(x) = u(x + a) define the translation operator, a ∈ R N .Then τ a u − u L 1 → 0 as |a| → 0.