An iterative method for non-autonomous nonlocal reaction-diffusion equations

The diffusion of a bacteria in a container or the study of the behaviour of a population which tends to move away from overpopulated areas are more accurate when the problem is modelled through a nonlocal diffusion equation (for more details see [9–12, 20]). In the last decades, many authors have been interested in studying a variant of the heat equation with a nonlocal diffusion term, namely

dudt−a(l(u))Δu=f,$$ \frac{du}{dt}-a(l(u))\Delta u=f, $$

where the function a is continuous and greater than a positive constant and l ∈ ℒ (L²(Ω),ℝ). One of the difficulties of this kind of problems is that the existence of a Lyapunov structure is not always guaranteed, only in particular situations (for more details cf. [15, 16], for p-laplacian see [13]).

Despite the limitations, Chipot and his coworkers have dealt with this type of equations in the last decades. Namely, they have been interested in studying the asymptotic behaviour of the solutions. In [8,10,11], assuming that function a is locally Lipschitz, they analyzed results which establish order relationships between the solution of the evolution problem and the corresponding stationary solutions. In fact, under suitable assumptions, they are able to prove the strong convergence of the solution of the evolution problem towards a stationary solution. To obtain their goal, they apply a wide range of techniques. For instance, dynamical systems are used in [11], Maximum Principle is applied in [10], a Lyapunov structure and minimizers of energy are used in [15], and relationships between the Lipschitz constant of the function a and the lower bound m are established in [14].

Besides some authors deal with equations with more than one nonlocal operator, for instance

dudt−A(l(u))u+a0(l(u))u=f,$$ \frac{du}{dt}-A(l(u))u+a_0(l(u))u=f, $$

being A=Σi,j=1N∂∂xi(aij(x)∂∂xj)$A=\sum_{i,j=1}^N{\frac{\partial}{\partial x_i}\left(a_{ij}(x)\frac{\partial}{\partial x_j}\right)}$ a uniformly elliptic operator (cf. for more details [7]). Issues as existence and uniqueness of weak solution and stationary solution, and the exponential decay of the evolution problem towards a stationary solution are analyzed. In addition, in a simpler framework

dudt−a(l(u))Δu+u=f,$$ \frac{du}{dt}-a(l(u))\Delta u+u=f, $$

Chipot and Chang [7] also prove results which establish relationships between weak solutions and stationary solutions similarly to the cited above, and the strong convergence of the weak solution towards a stationary solution (cf. [7, Lemma 4.5] and [7, Theorem 4.2] respectively).

Moreover, in this setting where the forcing term is linear, there exist some partial results concerning the existence of global attractors for autonomous nonlocal problems developed by Lovat [20] and Andami Ovono [1].

Other authors have been interested in including nonlinear and time-dependent forcing terms and have analysed the non-autonomous nonlocal diffusion problem

dudt−aluΔu=fu+htinΩ×τ,∞,u=0on∂Ω×τ,∞,ux,τ=uτxinΩ,$$ \left\{ \begin{array}{*{20}{l}} {\frac{{du}}{{dt}} - a\left( {l\left( u \right)} \right){\rm{\Delta }}u = f\left( u \right) + h\left( t \right)\quad \;\:{\rm{in\; \Omega }} \times \left( {\tau ,\infty } \right),}\\ {u = 0\quad \;\text{on}\;\partial {\rm{\Omega }} \times \left( {\tau ,\infty } \right),}\\ {u\left( {x,\tau } \right) = {u_\tau }\left( x \right)\quad \;\:{\rm{in\; \Omega }},} \end{array}\right. $$(1)

where

a∈C(ℝ;[m,∞)),$$ a \in C(;[m,\infty )), $$(2)

being m > 0, l a continuous linear form on L²(Ω), i.e.

l(u)=lg(u)=∫Ωg(x)u(x)dxfor some g∈L2Ω,$$ \begin{array}{*{20}{c}} {l(u) = {l_g}(u) = {\int _\Omega }g(x)u(x)dx\quad }&{{\text{for some }}g \in {L^2}\left( {\rm{\Omega }} \right)} \end{array}, $$(3)

u_τ ∈ L²(Ω) and h∈Lloc2(ℝ;H−1(Ω)).$h\in{L^2_{loc}(\mathbb{R};H^{-1}(\Omega))}$. When the functions a and f are also globally Lipschitz, the existence and uniqueness of weak solution to (1) have been proved in [21] by Menezes making use of fixed point arguments. Later, this result is proved in [3] via the compactness method when a is locally Lipschitz and f is just continuous, sublinear and fulfils for some η > 0

(f(s)−f(r))(s−r)≤η(s−r)2∀s,r∈ℝ.$$ (f(s)-f(r))(s-r)\leq{\eta(s-r)^2\quad\forall{s,r\in\mathbb{R}.}} $$(4)

There exist also results concerning the existence of pullback attractors in L²(Ω) and the upper semicontinuity property of attractors in the multi-valued framework for a variation of (1) with perturbations (see [4] for more details). In this case, the function f ∈ C(ℝ) fulfils

−κ−α1|s|p≤f(s)s≤κ−α2|s|p∀s∈ℝ,$$ -\kappa-\alpha_1|s|^p\leq{f(s)s}\leq{\kappa-\alpha_2|s|^p}\quad\forall{s}\in\mathbb{R}, $$(5)

where α₁, α₂, and κ are positive constants and p ≥ 2. Observe that from (5), we can deduce that there exists β > 0 such that

|f(s)|≤β(|s|p−1+1)∀s∈ℝ.$$ |f(s)|\leq{\beta(|s|^{p-1}+1)\quad\forall{s}\in\mathbb{R}.} $$(6)

Interested in improving the assumptions made in [3], there exist recent results that show, via the compactness method, the existence and uniqueness of a weak solution to (1) when the function f ∈ C(ℝ) satisfies (4) and (5) (see [6] for more details).

The goal of this paper is to show the existence of weak solutions to (1) on a bounded open set Ω ⊂ ℝ^N under assumptions (2)–(5), via the monotonicity method (cf. [19, Chapitre 2]), together with an iterative procedure and compactness arguments. Although the idea of using iterations to deal with an equation with nonlocal diffusion was applied in the proof of [17, Theorem 1.1], as far as we know, there are no references that combine monotonicity, iterations and compactness, to prove existence results in the nonlocal setting. In addition, we would like to point out that although making use of compactness by translation the existence of weak solutions to (1) can be proved, due to the several difficulties existing with this type of problems we consider that it is interesting to be able to attack the equation in so many different ways as possible (in order to try to apply to new extensions). In particular here we combine iterations, compactness arguments and monotonicity as mentioned before. Observe that this new method allows to use the well-known monotonicity method for solving PDEs even in the nonlocal framework and for general linear operators (see Conclusions and final comments at the end of the manuscript for more details).

The structure of the paper is as follows. In Section 2 (and just for completeness to the reader), we recall the monotonicity method for solving nonlinear PDEs. After that, in Section 3, we prove the existence of weak solutions for a non-autonomous nonlocal reaction-diffusion problem combining the previous results from Section 2 with compactness arguments and iterations. To conclude, we provide some comments about how to extend this procedure to some other linear operators.

Let us introduce the notation used throughout the paper. As usual, the inner product in L²(Ω) is denoted by (·,·) and by | · | the norm associated to it. The inner product in H01(Ω)$H^1_0(\Omega)$, which is given by the product in (L²(Ω))^N of the gradients, is represented by ((·,·)) and by || · || its associated norm. By 〈·,·〉, we represent the duality product between H⁻¹(Ω) and H01(Ω)$H^1_0(\Omega)$, and by ||·|| the norm in H⁻¹(Ω). We identify L²(Ω) with its dual, and therefore we have the chain of compact and dense embeddings H01(Ω)⊂L2(Ω)⊂H−1(Ω)$H^1_0(\Omega)\subset{L^2(\Omega)}\subset{H^{-1}(\Omega)}$. Observe that as a result of the previous identification, we can abuse of the notation considering l ∈ L²(Ω) but continue denoting (l,u) as l(u). The duality product between L^p(Ω) and L^q(Ω) (where q is the conjugate exponent of p) will be denoted by (·,·), and the norm in L^s(Ω) will be represented by ||·||_L^s(Ω) with s ≥ 1. We also denote by 〈·,·〉 the duality product between H⁻¹(Ω) + L^q(Ω) and H01(Ω)∩Lp(Ω)$H^1_0(\Omega)\cap{L^p(\Omega)}$.

In this section we provide a short summary about the requirements to apply the monotonicity method for solving nonlinear PDEs (see [19, Chapitre 2] for more details).

Consider a separable Hilbert space H, whose norm is denoted by |·|. Moreover, suppose given V_i, i = 1,...,m, with m ≥ 1, separable and reflexive Banach spaces, such that ∪i=1mVi⊂H,∩i=1mVi$\bigcup_{i=1}^m{V_i}\subset{H}, \bigcap_{i=1}^m{V_i}$ is dense in H, and V_i ⊂ H with continuous injection for all i = 1,...,m.

We denote by || · ||_i and || · ||_*i the norms in V_i and Vi′$V_i'$ respectively (i = 1,...,m). By V we represent the space ∩i=1mVi$\bigcap_{i=1}^m{V_i}$. In addition, 〈·,·〉 denotes the duality product between Vi′$V'_i$ and V_i for all i = 1,...,m. Finally, H is identified with its topological dual H^′ using the Riesz theorem.

Consider T ∈ (τ, ∞) fixed and let Bi:(τ,T)×Vi→Vi′$B_i:(\tau,T)\times{V_i}\to{V'_i}$ be, for i = 1,...,m, operators, in general nonlinear, such that

A1) The mapping (τ,T)∋t↦Bi(t,v)∈Vi′$(\tau,T)\ni t\mapsto{B_i(t,v)}\in{V'_i}$ is measurable for each v ∈ V.

Suppose also that there exist 1 < p_i < ∞, i = 1,...,m, at least one of them greater than or equal to 2, constants c > 0, α > 0 and λ ≥ 0, and a nonnegative function C ∈ L¹(τ, T), such that for all t ∈ (τ, T) it satisfies

A3) B_i(τ, ·) is bounded in Vi′$V_i'$, i.e.

||Bi(t,v)||*i≤c(1+||v||ipi−1)∀v∈Vi.$$ {||{B_i}(t,v)|{|_{*i}} \le c(1 + ||v||_i^{{p_i} - 1})} \quad {\forall v \in {V_i}.} $$

Now, we consider the problem

u∈⋂i=1mLpi(τ,T;Vi)∀T>τ,u′(t)+B(t,u(t))=h(t)in D′(τ,T;V′),u(τ)=uτ.$$\left\{\begin{array}{l} u\in\displaystyle{\bigcap_{i=1}^m{L^{p_i}(\tau,T;V_i)}}\quad\forall{T>\tau},\\[2ex] u'(t)+B(t,u(t))=h(t)\quad\mbox{in }\mathcal{D}'(\tau,T;V'),\\[2ex] u(\tau)=u_\tau. \end{array} \right.$$(8)

Then, we have the following result (see [19, Théorème 1.4, p. 168]). Observe that in the proof, it can be used any numerable family formed by linearly independent elements such that the vector space generated by this family is dense in V.

In this section we will prove the existence of a weak solution to (1) using iterations, compactness arguments and the monotonicity method for solving nonlinear PDEs. Due to the presence of the nonlocal operator in the diffusion term, the monotonicity method recalled in Section 2 does not seem possible to be applied directly to problem (1). However we first apply this method to a non-autonomous reaction-diffusion equation in which the diffusion term has a viscosity which depends on time, but it does not depend on the solution. Then through iterations and appropriate estimates, we can prove the existence and uniqueness of a weak solution to (1) using compactness arguments.

For each n ≥ 1, we denote by uⁿ the weak solution to

(Pn){u∈L2(τ,T;H01(Ω))∩Lp(τ,T;Lp(Ω))∩C([τ,T];L2(Ω)),ddt(u(t),v)+a(l(un−1(t)))((u(t),v))=(f(u(t)),v)+〈h(t),v〉,u(τ)=uτ,$$ (P_n) \left\{\begin{array}{l} u\in{L^2(\tau,T;H^1_0(\Omega))\cap{L^p(\tau,T;L^p(\Omega))}\cap{C([\tau,T];L^2(\Omega))}},\\[2ex] \displaystyle\frac{d}{dt}(u(t),v)+a(l(u^{n-1}(t)))((u(t),v))=(f(u(t)),v)+\langle h(t),v\rangle,\\[2ex] u(\tau)=u_\tau, \end{array} \right. $$

with u⁰ ≡ 0 and uⁿ is the solution to (P_n) if n ≥ 1, where the equation in (P_n) must be understood in the sense of the distributions 𝒟^′(τ, ∞) for all v∈H01(Ω)∩Lp(Ω)$v\in{H^1_0(\Omega)\cap{L^p(\Omega)}}.$.

Integrating between τ and t ∈ [τ, T], we deduce that {uⁿ} is bounded in L∞(τ,T;L2(Ω))∩L2(τ,T;H01(Ω))∩Lp(τ,T;H01(Ω))$L^\infty(\tau,T;L^2(\Omega))\cap{L^2(\tau,T;H^1_0(\Omega))}\cap{L^p(\tau,T;L^p(\Omega))}$. From this, taking into account that each uⁿ ∈ C([τ, T];L²(Ω)), we deduce that there exists a positive constant Ĉ such that

|un(t)≤C^∀t∈[τ,T]∀n≥1.$$\eqalign{ {|{u^n}\left( t \right) \le \hat C}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\forall t \in \left[ {\tau ,T} \right]} \end{array}}&{\forall n \ge 1.} \end{array}} }$$

Then, using that the function a ∈ C(ℝ;ℝ₊) and l ∈ L²(Ω), there exists a positive constant M_Ĉ such that

a(l(un−1(t)))≤MC^∀t∈[τ,T]∀n≥1.$$\eqalign{ a(l(u^{n-1}(t)))\leq{M_{\widehat C}}\quad\forall{t}\in[\tau,T]\quad\forall n\geq{1}. }$$

Therefore, it fulfils that {−a(l(uⁿ⁻¹))Δuⁿ} is bounded in L²(τ, T;H⁻¹(Ω)). Moveover, making use of the boundedness of {u_n} in L^p(τ, T;L^p(Ω)), we obtain that the sequence {f(u_n)} is bounded in L^q(τ, T;L^q(Ω)).

Now, since it satisfies for all n ≥ 1

dundt−a(l(un−1))Δun=f(un)+hinD′(τ,T;H−1(Ω)+Lq(Ω)),$$\eqalign{ \frac{du^n}{dt}-a(l(u^{n-1}))\Delta u^n=f(u^n)+h\quad\mbox{in }\mathcal{D}'(\tau,T;H^{-1}(\Omega)+L^q(\Omega)), }$$

the sequence {(uⁿ)^′} is bounded in L²(τ, T;H⁻¹(Ω)) + L^q(τ, T;L^q(Ω)). Therefore, there exist a subsequence of {uⁿ} (relabeled the same), a function u∈L∞(τ,T;L2(Ω))∩L2(τ,T;H01(Ω))∩Lp(τ,T;Lp(Ω))$u\in L^\infty(\tau,T;L^2(\Omega)) \cap{L^2(\tau,T;H^1_0(\Omega))}\cap{L^p(\tau,T;L^p(\Omega))}$ with u^′ ∈ L²(τ, T;H⁻¹(Ω)) + L^q(τ, T;L^q(Ω)), ξ₁ ∈ L^q(τ, T;L^q(Ω)), and ξ2∈L2(τ,T;H01(Ω))$\xi_2\in{L^2(\tau,T;H^1_0(\Omega))}$, such that

{un⇀*uWeakly-star inL∞(τ,T;L2(Ω)),un⇀uWeakly-star inL2(τ,T;H01(Ω)),un⇀uWeakly-star inLp(τ,T;Lp(Ω)),un⇀ustrongly inL2(τ,T;L2(Ω)),(un)′⇀u′Weakly-star inL2(τ,T;H−1(Ω))+Lq(τ,T;Lq(Ω)),f(un)⇀ξ1Weakly-star inLq(τ,T;Lq(Ω)),a(l(un−1))un⇀ξ2Weakly-star inL2(τ,T;H01(Ω)).$$\eqalign{ \left\{\begin{aligned} u^n&\stackrel{*}\rightharpoonup u\quad\mbox{weakly-star in }L^\infty(\tau,T;L^2(\Omega)),\\ u^n&\rightharpoonup {u}\quad\mbox{weakly in }L^2(\tau,T;H^1_0(\Omega)),\\ u^n&\rightharpoonup u\quad\mbox{weakly in }L^p(\tau,T;L^p(\Omega)),\\ u^n&\to u\quad\mbox{strongly in }L^2(\tau,T;L^2(\Omega)),\\ (u^n)'&\rightharpoonup{u'}\quad\mbox{weakly in }L^2(\tau,T;H^{-1}(\Omega))+L^q(\tau,T;L^q(\Omega)),\\ f(u^n)&\rightharpoonup{\xi_1}\quad\mbox{weakly in $L^q(\tau,T;L^q(\Omega)),$}\\ a(l(u^{n-1}))u^n&\rightharpoonup{\xi_2\quad\mbox{weakly in }L^2(\tau,T;H_0^{1}(\Omega))}. \end{aligned}\right. }$$(9)

Now, we will prove that ξ₁ = f(u) and ξ₂ = a(l(u))u. From (9), we deduce

un(x,t)→u(x,t)strongly∀(x,t)∈(Ω×(τ,T))\N1,$$\eqalign{ {{u^n}\left( {x,t} \right) \to u\left( {x,t} \right)}&{\begin{array}{*{20}{c}} {{\rm{strongly}}}&{\forall \left( {x,t} \right) \in \left( {{\rm{\Omega }} \times \left( {\tau ,T} \right)} \right)\backslash {N_1},} \end{array}} }$$(10)

un(t)→u(,t)strongly inL2(Ω)∀t∈(τ,T)\N2,$$\eqalign{ {{u^n}\left( t \right) \to u\left( {,t} \right)}&{\begin{array}{*{20}{c}} {{\text{strongly in }}{L^2}\left( {\rm{\Omega }} \right)}&{\forall t \in \left( {\tau ,T} \right)\backslash {N_2},} \end{array}} }$$(11)

where N₁ is a null set in ℝ^N+1 and N₂ is a null set in ℝ.

Then, since the sequence {f(uⁿ)} is bounded in L^q(τ, T;L^q(Ω)), f ∈ C(ℝ) and (10) holds, it fulfils that ξ₁ = f(u), thanks to [19, Lemme 1.3, p. 12].

Finally, we will prove ξ₂ = a(l(u))u. Since a ∈ C(ℝ;ℝ₊), l ∈ L²(Ω) and (11) holds, we have

a(l(un−1(t)))→a(l(u(t)))∀t∈(τ,T)\N2.$$\eqalign{ {a\left( {l\left( {{u^{n - 1}}\left( t \right)} \right)} \right) \to a\left( {l\left( {u\left( t \right)} \right)} \right)}&{\forall t \in \left( {\tau ,T} \right)\backslash {N_2}.} }$$

Therefore,

a(l(un−1(t)))un(x,t)→a(l(u(t)))u(x,t)∀(x,t)∈(Ω×(τ,T))\(N1∪(Ω×N2)),$$\eqalign{ {a\left( {l\left( {{u^{n - 1}}\left( t \right)} \right)} \right){u^n}\left( {x,t} \right) \to a\left( {l\left( {u\left( t \right)} \right)} \right)u\left( {x,t} \right)}&{\forall \left( {x,t} \right) \in ({\rm{\Omega }} \times \left( {\tau ,T} \right))\backslash ({N_1} \cup \left( {{\rm{\Omega }} \times {N_2}} \right)),} }$$

where N₁ ∪ (Ω × N₂) is a null set in ℝ^N+1. From this and the boundedness of {a(l(uⁿ⁻¹))uⁿ} in L2(τ,T;H01(Ω))$L^2(\tau,T;H^1_0(\Omega))$, it fulfils that ξ₂ = a(l(u))u, applying again [19, Lemme 1.3, p. 12].

Thereupon, to prove (7) for all v∈H01(Ω)∩Lp(Ω)$ v\in{H^1_0(\Omega)\cap{L^p(\Omega)}} $, we fix T > τ and φ ∈ 𝒟(τ, T). Since uⁿ is a solution to (P_n), it satisfies for all n ≥ 1

∫τT(un(t),v)φ′(t)dt+∫τTa(l(un−1(t)))〈−Δun(t),v〉φ(t)dt=∫τT〈f(un(t))+h(t),v〉φ(t)dt.$$\eqalign{ \int_\tau^T{(u^n(t),v)\varphi'(t)}dt+\int_\tau^T{a(l(u^{n-1}(t)))\langle -\Delta u^n(t),v\rangle\varphi(t)}dt=\int_\tau^T{\langle f(u^n(t))+h(t),v\rangle \varphi(t)}dt. }$$

Taking limit when n → ∞ in the previous expression and making use of (9), (7) holds.

Finally, to prove the existence of a weak solution to (1), we only need to check that u(τ) = u_τ. Observe that this equality makes complete sense since u ∈ C([τ, T];L²(Ω)) (cf. [18, Théorème 2, p. 575]). To do this, consider fixed v∈H01(Ω)∩Lp(Ω)$v\in{H^1_0(\Omega)\cap L^p(\Omega)}$ and φ ∈ H¹(τ, T), with φ(T) = 0 and φ(τ) ≠ 0. Since uⁿ is a weak solution to (P_n), it fulfils

ddt(un(t),v)+a(l(un−1(t)))((un(t),v))=(f(un(t)),v)+〈h(t),v〉a.e.t∈(τ,T).$$\eqalign{ {\frac{d}{{dt}}\left( {{u^n}\left( t \right),v} \right) + a\left( {l\left( {{u^{n - 1}}\left( t \right)} \right)} \right)\left( {\left( {{u^n}\left( t \right),v} \right)} \right) = \left( {f\left( {{u^n}\left( t \right)} \right),v} \right) + \langle h\left( t \right),v\rangle }&{{\rm{a}}{\rm{.e}}{\rm{.}}t \in \left( {\tau ,T} \right).} }$$

Now, multiplying by φ in the previous expression and integrating between τ and T, we have

−(uτ,v)φ(τ)+∫τT(un(t),v)φ′(t)dt+∫τTa(l(un−1(t)))((un(t),v))φ(t)dt=∫τT(f(un(t)),v)φ(t)dt+∫τT〈h(t),v〉φ(t)dt.$$\eqalign{ {}&{ - \left( {{u_\tau },v} \right)\varphi \left( \tau \right) + \smallint _\tau ^T\left( {{u^n}\left( t \right),v} \right)\varphi '\left( t \right)dt + \smallint _\tau ^Ta\left( {l\left( {{u^{n - 1}}\left( t \right)} \right)} \right)\left( {\left( {{u^n}\left( t \right),v} \right)} \right)\varphi \left( t \right)dt}\\ = &{\smallint _\tau ^T\left( {f\left( {{u^n}\left( t \right)} \right),v} \right)\varphi \left( t \right)dt + \smallint _\tau ^T\langle h\left( t \right),v\rangle \varphi \left( t \right)dt.} }$$

Thereupon, taking limit in n, we obtain

−(uτ,v)φ(τ)+∫τT(u(t),v)φ′(t)dt+∫τTa(l(u(t)))((u(t),v))φ(t)dt=∫τT(f(u(t)),v)φ(t)dt+∫τT〈h(t),v〉φ(t)dt.$$\eqalign{ {}&{ - \left( {{u_\tau },v} \right)\varphi \left( \tau \right) + \smallint _\tau ^T\left( {u\left( t \right),v} \right)\varphi '\left( t \right)dt + \smallint _\tau ^Ta\left( {l\left( {u\left( t \right)} \right)} \right)\left( {\left( {u\left( t \right),v} \right)} \right)\varphi \left( t \right)dt}\\ = &{\smallint _\tau ^T\left( {f\left( {u\left( t \right)} \right),v} \right)\varphi \left( t \right)dt + \smallint _\tau ^T\langle h\left( t \right),v\rangle \varphi \left( t \right)dt.} }$$

Otherwise, we deduce from (7)

−(u(τ),v)φ(τ)+∫τT(u(t),v)φ′(t)dt+∫τTa(l(u(t)))((u(t),v))φ(t)dt=∫τT(f(u(t)),v)φ(t)dt+∫τT〈h(t),v〉φ(t)dt.$$\eqalign{ {}&{ - \left( {u\left( \tau \right),v} \right)\varphi \left( \tau \right) + \smallint _\tau ^T\left( {u\left( t \right),v} \right)\varphi '\left( t \right)dt + \smallint _\tau ^Ta\left( {l\left( {u\left( t \right)} \right)} \right)\left( {\left( {u\left( t \right),v} \right)} \right)\varphi \left( t \right)dt}\\ = &{\smallint _\tau ^T\left( {f\left( {u\left( t \right)} \right),v} \right)\varphi \left( t \right)dt + \smallint _\tau ^T\langle h\left( t \right),v\rangle \varphi \left( t \right)dt.} }$$

Comparing these last two equalities we deduce that (u(τ), v)φ(τ) = (u_τ, v)φ(τ). Finally, since φ(τ) ≠ 0 and H01(Ω)∩Lp(Ω)$ H^1_0(\Omega)\cap{L^p(\Omega)} $ is dense in L²(Ω), the equality u(τ) = u_τ holds.

Since u₁, u₂ ∈ C([τ, T];L²(Ω)), therefore {u_i(t)}_i=1,2 ⊂ S for all t ∈ [τ, T], where S is a bounded set of L²(Ω). Moreover, it satisfies {l(u_i(t))}_i=1,2 ⊂ [−R, R] for all t ∈ [τ, T], for some R > 0, since l ∈ L²(Ω). Therefore, using (2), (4) and the locally Lipschitz continuity of the function a, we have

ddt|u1(t)−u2(t)|2≤(La(R))2|l|2||u2(t)||2+4ηm2m|u1(t)−u2(t)|2a.e.t∈(τ,T),$$\eqalign{ {\frac{d}{{dt}}|{u_1}\left( t \right) - {u_2}\left( t \right){|^2} \le \frac{{{{({L_a}\left( R \right))}^2}\left| {l{|^2}} \right|\left| {{u_2}\left( t \right)} \right|{|^2} + 4\eta m}}{{2m}}|{u_1}\left( t \right) - u2\left( t \right){|^2}}&{{\rm{a}}{\rm{.e}}{\rm{.}}t \in \left( {\tau ,T} \right),} }$$

where L_a(R) is the Lipschitz constant of the function a in [−R, R]. Then, both statements, uniqueness and continuity w.r.t. initial data, hold.

We have presented a procedure to obtain existence of solution for a nonlocal reaction-difussion problem through a combination of different methods, namely monotonicity, iterations and compactness arguments. To go further on the study of the long-time behaviour and regularity issues related to (1) and variations we refer to [3–6] and the cited references in Section 1. As referred in that paragraph, these results ensure the global-intime existence of solutions to some nonlocal models settled in Biology and Physics among other areas.

It is delicate how to generalise the method established in this paper to other nonlocal problems with monotone operators but nonlinear (see e.g. [5]). Now, we provide some extensions for linear operators.

The previous procedure can be applied to equations with more general linear diffusion terms with divergence form like a(l(u))Au, where A=Σi,j=1N∂∂xi(bij(x)∂∂xj)$ A=\sum_{i,j=1}^N{\frac{\partial}{\partial x_i}\left(b_{ij}(x)\frac{\partial}{\partial x_j}\right)} $, with b_ij ∈ L^∞(Ω) for all i, j = 1,...,N and Σi,j=1Nbij(x)ξiξj≥ζ|ξ|ℝN2$ \sum_{i,j=1}^N b_{ij}(x)\xi_i\xi_j\geq\zeta|\xi|^2_{\mathbb{R}^N} $, where ζ > 0. In fact, it is possible to apply this procedure to nonlocal equations given by a sum of a finite family of terms under the above assumptions, namely

dudt−Σk=1mak(lk(u))Aku=Σk=1m(fk(u)+hk(t)),$$\eqalign{ \frac{du}{dt}-\sum_{k=1}^m a_k(l_k(u))A_ku=\sum_{k=1}^m(f_k(u)+h_k(t)), }$$

where the nonlocal operator a_k(l_k(u)) can be different for each k, f_k fulfils

−κ−α1|s|pk≤fk(s)s≤κ−α2|s|pk∀s∈ℝ,$$\eqalign{ -\kappa-\alpha_1|s|^{p_k}\leq f_k(s)s\leq \kappa-\alpha_2|s|^{p_k}\quad\forall{s\in\mathbb{R}}, }$$

and Σ_k h_k belongs to Lloc2(ℝ;H−1(Ω))+Σk=1mLlocpk′(ℝ;Lpk′(Ω))$ {L^2_{loc}(\mathbb{R};H^{-1}(\Omega))}+\sum_{k=1}^mL^{p'_k}_{loc}(\mathbb{R};L^{p'_k}(\Omega)) $.