Homogenization of Smoluchowski-type equations with transmission boundary conditions
-
Bruno Franchi
Abstract
In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.
1 Introduction
Let us consider a bounded open set Ω in
and D ϵ = D\G ϵ , as well as
Thus, the geometric structure of the domain in
Geometry of the problem.
Throughout this paper, ϵ will denote the general term of a sequence of positive reals which converges to zero. Let us introduce two nonnegative vector-valued functions,
For 1 ≤ m ≤ M we have:
where
and
In the system of equations above, ν
ϵ
is the outer normal on Γ
ϵ
with respect to Ω
ϵ
; f
ϵ
(t, x, z) ∈ C
1([0, T] × Π
ϵ
), c
i
(x, z) ∈ L
∞(Γ
ϵ
)(i = 1, …, M) and
Our main statement shows that it is possible to homogenize the system of Equations (1.1) as ϵ → 0.
Theorem 1.1.
Let
For 1 ≤ m ≤ M we have:
where
In (1.8) and (1.9), A is a matrix with constant coefficients defined by
with
appearing in the limiting function
Furthermore, for 1 ≤ m ≤ M we have:
where
and
1.1 Motivation
The system of equations in (1.1), known under the generic name of discrete Smoluchowski equations with diffusion, is meant to model the aggregation and diffusion of the pathological tau protein in the brain, a process associated with the development of a large variety of cerebral diseases called tauopathies. Indeed, pathological accumulations of hyperphosphorylated tau protein aggregates, known as neurofibrillary tangles (NFTs), are detected in several neurodegenerative tauopathies, including Alzheimer’s disease (AD) [2], [3], [4], [5]. Tau is a highly soluble, natively unfolded protein which is predominantly located in the axons of neurons of the central nervous system. Here, its physiological function is to support assembly and stabilization of axonal microtubules. Under pathological conditions, tau can assume abnormal conformations, due to two transformations: hyperphosphorylation and misfolding. In particular, hyperphosphorylation has a negative impact on the biological function of tau proteins, since it inhibits the binding to microtubules, compromising their stabilization and axonal transport, and promotes self-aggregation. Thus, misfolded tau monomers constitute the building unit for the formation of oligomers, which in turn lead to highly structured and insoluble fibrils. For many years, cell autonomous mechanisms were believed to be responsible for the evolution of neurodegenerative diseases, implying that the same aggregation events occur independently in different brain cells. However, accumulating evidence now demonstrates that the progression of tau pathology reflects cell-to-cell propagation of the disease, achieved through the release of tau into the extracellular space and the uptake by surrounding healthy neurons [6], [7]. Extracellular tau then seeds physiological tau in the recipient cells propagating the pathological process through neural pathways made by bundles of axons (called tracts). This mechanism, often referred to as ‘prion-like’ propagation of tau pathology, has led to the idea that extracellular tau could be a novel therapeutic target to halt the spread of the disease [8]. For this reason, in the present work, we focus on a model which describes, in a simplified way, the tau diffusion along tracts starting from the microscopic release mechanism.
In this context, the set Π
ϵ
represents a bundle of axons in the white matter (a tract), while the domain Ω
ϵ
indicates the extracellular region filled by cerebrospinal fluid. The variables
1.2 Background of this work
There is a large literature related to the use of the Smoluchowski equation in various physical contexts (e.g., Refs. [9], [10], [11], [12], [13], [14]), but only a few works concerning its application in the biomedical field [15], [16]. Recently, the important role of the Smoluchowski equations in modelling at different scales the evolution of neurodegenerative diseases, such as AD, has been investigated in Refs. [15], [17]–[23]. In fact, the present work is part of a broader research effort carried on by various groups of researchers with diverse collaborations on mathematical models of the progression of AD, and represents an initial bridge between microscopic models of tau diffusion developed in biology and macroscopic mathematical models based on graph theory. To this end, the homogenization technique, introduced by the mathematicians in the seventies to carry out a sort of averaging process on the solutions of partial differential equations with rapidly varying coefficients or describing media with microstructures, has been applied [24], [25], [26], [27], [28], [29], [30].
It is nowadays generally accepted that tau protein, in synergetic combination with another protein, the so-called β-amyloid peptide, plays a key role in the development of AD (see Ref. [31]). We refer for instance to Ref. [15] for a discussion on macroscopic mathematical modeling of this interaction. Here, our interest is focused on the tau protein that diffuses through the neural pathway, whereas we ignore deliberately the action of the β-amyloid.
Unlike the approach proposed for instance in Refs. [15], [23], where the modeling of tau coagulation-diffusion processes has been carried out on a large scale, that is the scale of the neural network characterized by the connectivity of different regions through bundles of axons (tracts), in this paper only the mesoscopic dynamics within a portion of tract has been investigated. Starting from the derivation of model equations valid at the microscale, by using the so-called two-scale homogenization technique, we have proved that the solution two-scale converges to the solution of a macromodel asymptotically consistent with the original one. The notion of two-scale limit has been first introduced by Nguetseng [32] and Allaire [33] in the deterministic periodic setting, and later generalized to the stochastic framework by Zhikov and Piatnitsky [34]. Unlike other homogenization techniques (see Refs. [26], [27] for a review), the two-scale convergence method is self-contained in that, in a single step, one can derive the homogenized equations and prove the convergence of the sequence of solutions to the problem at hand.
To stress the novelty of the present paper, it is worth noting that two-scale homogenization techniques have been already used by the authors to pass from microscopic to macroscopic model of the diffusion of toxic proteins in the cerebral parenchyma affected by AD, but in completely different biological perspectives and geometries. Indeed, in Refs. [19], [20], the authors aimed to describe production, aggregation and diffusion of β-amyloid peptide in the cerebral tissue (macroscopic scale), a process associated with the development of AD, starting from the derivation of a model at the single neuron level (microscopic scale). Thus, the present paper differs from our previous works both for the biological meaning and, consequently, for the geometry of the problem and the boundary conditions introduced in order to take into account the peculiarities of tau protein propagation.
There is a large literature devoted to the study of transmission boundary conditions somehow similar to those of (1.1), especially in the framework of porous media [35]. The main results in this respect, which are relevant to our work, can be found in Refs. [36], [37], [38]. In particular, in Ref. [36], it is assumed that the porous medium is composed of periodically arranged cubic cells of size ϵ, split up into a solid part (a ball surrounded by semi-permeable membranes) and a fluid part. In this setting, the diffusion and reactions of chemical species in the fluid and in the solid part are studied, while transmission boundary conditions are imposed on the interface. The same geometry is considered also in Ref. [37], where deposition effects under the influence of thermal gradients are analyzed. The model takes into account the motion of populations of colloidal particles dissolved in the water interacting together via Smoluchowski coagulation terms. The colloidal matter cannot penetrate the solid grain boundary, but it deposits there. This process is again described by transmission boundary conditions. A domain decomposed into long cylindrical cavities periodically distributed has been defined in Ref. [38] to model a porous medium consisting of a fluid part and solid bars. Chemical substances, dissolved in the fluid, are transported by diffusion and adsorbed on the surface of the bars (through transmission boundary conditions) where chemical reactions take place.
1.3 Outline of the paper
The paper is organized as follows. In Section 2 we prove the existence of weak solutions to the system of Smoluchowski-type Equations (1.1), while Sections 3 and 4 are devoted to the proof of their positivity, boundedness and uniqueness, respectively. A priori estimates on the derivatives of the solutions are also obtained and reported at the end of Section 3. In Section 5 we study the convergence of the homogenization process and prove our main result concerning the two-scale limit of the solutions to the set of Equations (1.1). Finally, in the Appendices we recall some basic Theorems related to functional analysis and on the two-scale convergence method.
2 Existence of solutions
Let us consider the following truncation of the nonlinear terms in the Smoluchowski-type Equations (1.1) for 1 ≤ i ≤ M [37]:
where
with
Definition 2.1.
The functions
If 1 ≤ i ≤ M:
for all ψ i ∈ H 1(Π ϵ ), and
for all ϕ
i
∈ H
1(Ω
ϵ
), along with the initial conditions
Remark 2.1.
In Eqs. (2.4) and (2.5) the integrals over the boundary Γ
ϵ
are well defined thanks to the regularity of the functions
Lemma 2.1.
For a given small ϵ > 0, the system (1.1) has a solution
Proof.
Let {ξ
j
} be an orthonormal basis of H
1(Π
ϵ
) and {η
j
} of H
1(Ω
ϵ
). We denote by
for all t ∈ [0, T], (x, z) ∈ Π ϵ and
for all t ∈ [0, T], (x, z) ∈ Ω ϵ . Let now i = 1.
Since {ξ
j
} is an orthonormal basis of H
1(Π
ϵ
), for each
so that
Likewise, for each
so that
To derive the coefficients of the Galerkin approximations, we impose that the functions
By testing Eq. (2.10) with ψ 1 = ξ k , we get
Hence, for k ∈ {1, …, n}:
where the coefficients A 1jk are defined by
and
We choose now
By testing Eq. (2.15) with ϕ 1 = η k , we get
Hence, for k ∈ {1, …, n}:
where the coefficients B 1jk are defined by
and
Equations (2.12) and (2.17) represent a system of 2n ordinary differential equations for the coefficients
Since the left-hand side of this system of ordinary differential equations is linear and the right-hand side has a sublinear growth (thanks to the Lipschitz property of the functions), one can conclude that the Cauchy problem (2.12), (2.17), and (2.20) has a unique solution extended to the whole interval [0, T] (see the generalization of the Picard–Lindelöf theorem in Ref. [39], Theorem 5.1, p. 156). Moreover,
Let
One has:
where the coefficients (c jk ) and (d jk ) are given by
for j, k = 1, …, n. Hence, we get
The same conclusions can be drawn also when 1 < i ≤ M by applying exactly the arguments considered above.
2.1 Uniform estimates
Let us now prove uniform estimates in n for
Since the first term on the right-hand side is always negative due to the truncation of the coagulation terms (see Eq. (2.3)), one obtains:
where we have applied the Hölder inequality to the last term on the right-hand side of Eq. (2.25). Since the function f ϵ (t, x, z) is bounded in L 2([0, T] × Π ϵ ), Eq. (2.26) reads:
where C
f
is a positive constant. By testing now Eq. (2.15) with
Since again the term on the right-hand side is negative, we conclude:
Adding the inequalities (2.27) and (2.29), it follows that
Let us estimate the second term on the right-hand side of (2.30):
Applying the generalized interpolation-trace inequality (A.7) in Appendix A to each term inside the round brackets, one has:
where η is a small positive constant. If we take into account the estimate (2.32), the inequality (2.30) reads:
If one chooses
Integrating Eq. (2.34) over [0, t] with t ∈ [0, T], we get
Since the sequences
Applying Gronwalls’s inequality, we obtain
Therefore, given ϵ ∈ [0, 1], η is fixed and for t ∈ [0, T] we get:
where C 3 is a positive constant independent of n and ϵ.
By testing Eqs. (2.4) and (2.5), in the case 1 < i ≤ M, with
where C and
Thus, we can conclude that:
Let us now derive uniform estimates in n for
where
By taking into account the Hölder and Young inequalities, Eq. (2.41) becomes
Exploiting the truncation of the coagulation terms and choosing
where
Let us now test Eq. (2.15) with the function
By applying once again the Hölder and Young inequalities to the right-hand side as above, and exploiting Eq. (2.3), we end up with the following expression:
where
where C is a positive constant. If we decompose now the function
Integrating over [0, t] with t ∈ [0, T], we deduce:
Hence, taking into account that the last term on the right-hand side of Eq. (2.48) is negative, one has
Since the sequences
where
In the case 1 < i ≤ M, by testing Eqs. (2.4) and (2.5) with
due to the boundedness of the coagulation terms given by (2.3). In Eqs. (2.51) and (2.52), C and
Thus, combining the estimates (2.38) and (2.50), one concludes that:
Proposition 2.1.
Since
as n → ∞. Since L 2([0, T]; H 1(Π ϵ )) ⊂ L 1([0, T]; H 1(Π ϵ )), formula (2.53) holds also for ψ ∈ L 2([0, T]; H 1(Π ϵ )).
In addition,
Finally, since in particular
by the Aubin–Lions–Simon theorem ([40], Theorem II.5.16, p. 102) (see Appendix B) we can infer that
An analogous proposition can be proved also for the sequence
Now, integrating Eq. (2.10) with respect to time and using as a test function ψ
1 = ϕ(t)ζ
1(x, z), with
where we denote by
Exploiting the convergence results stated in Proposition 2.1 we can pass to the limit as n → ∞ to obtain:
where we have taken into account that the term
By using the same arguments handled above for
It remains to show that the initial conditions hold. By the Aubin–Lions–Simon theorem it follows that
3 Positivity and boundedness of solutions
Lemma 3.1.
For a given small ϵ > 0, let
Proof.
In the case i = 1, let us test Eq. (2.4) with the function
Decomposing the function
Since the last but one term on the right-hand side is always zero and the last one is negative, one obtains:
Let us now take
Since the last term on the right-hand side is always zero, Eq. (3.4) yields:
Adding Eqs. (3.3) and (3.5) we obtain:
Let us now estimate the term on the right-hand side of Eq. (3.6):
where we denote by
Exploiting the Hölder and Young inequalities along with the generalized interpolation-trace inequality (A.7), Eq. (3.7) becomes
where η is a small constant. Finally from Eqs. (3.6) and (3.8) it follows that
If one chooses
Setting the initial conditions:
that is,
In the case 1 < i ≤ M, by testing Eqs. (2.4) and (2.5) with
Since in both Eqs. (3.12) and (3.13) the last but one term on the right-hand side is negative and the last one always zero, we obtain that also the functions
Let us now prove the boundedness of solutions. In the case i = 1, we test Eq. (2.4) with
Since the last term on the left-hand side is positive and the first term on the right-hand side is negative, Eq. (3.14) implies:
Taking into account that the last two terms on the left-hand side are positive and integrating over [0, t] with t ∈ [0, T], we obtain:
Hence,
In order to estimate the last term on the right-hand side of Eq. (3.17), it is now convenient to use Young’s inequality in the following form [41]:
with:
The last line above has been obtained by making the following approximation, which holds since, to finalize the proof, we will take the limit p → ∞:
Choosing η = p and using the inequality (3.19) in (3.17), we conclude that:
Applying Gronwall’s inequality it follows that:
And finally
where M
1 is a positive constant due to the boundedness of the initial condition
We test now Eq. (2.5) with
Decomposing the function
Since the last term on the right-hand side is negative, we conclude:
Let us now estimate the term on the right-hand side of (3.25):
if one chooses
where η is a small constant and
If we choose
and the Gronwall lemma gives:
for all t ∈ [0, T]. Since
and from Eq. (3.30) it follows:
In the case 1 < i ≤ M we proceed by induction and test Eq. (2.4) with
Since the last term on the left-hand side is positive and the first term on the right-hand side is negative, Eq. (3.32) reduces to:
Taking into account that the last two terms on the left-hand side are positive and integrating over [0, t] with t ∈ [0, T], we obtain:
Exploiting the boundedness of
where K
j
(1 ≤ j ≤ i − 1) are positive constants. In order to estimate the term on the right-hand side of Eq. (3.35), we use the Young inequality (3.18) with:
Choosing η = p and using the inequality (3.36) in (3.35), we obtain:
The Gronwall lemma applied to (3.37) leads to the estimate:
Hence,
where M i is a positive constant.
We test now Eq. (2.5) with
Decomposing the function
where K j (1 ≤ j ≤ i − 1) are positive constants. Since the last term on the left-hand side is positive and the one on the right-hand side is negative, Eq. (3.41) reduces to:
Let us now estimate the first term, I 1, on the right-hand side of Eq. (3.42):
where the last step in inequality (3.43) has been obtained decomposing the functions
The right-hand side of this inequality vanishes since
Therefore, Eq. (3.42) reduces to:
Integrating over [0, t] with t ∈ [0, T], and estimating the term on the right-hand side by using Young’s inequality in the form (3.18), with:
Choosing η = p and setting the initial conditions, it follows that:
Finally, the Gronwall Lemma leads to the estimate
Hence,
Therefore, since the positive part of the function
Lemma 3.2.
For a given small ϵ > 0, let
where 1 ≤ i ≤ M and
Proof.
In the case i = 1, let
For the case 1 < i ≤ M, the proof carries over verbatim, since also the functions
4 Uniqueness of solutions
Theorem 4.1.
Let
Then,
Proof.
From now on, we drop the index ϵ and we set
Then, in Eqs. (2.1) and (2.2), we choose
so that
and
Let us now write Eq. (2.4) for (u
i
, v
i
) and (
for all ψ i ∈ H 1(Π ϵ ).
Since the positive part function is 1-Lipschitz, the following estimate holds:
Thus
where C is a positive constant. Moreover, we have
since L
i
is a Lipschitz continuous function, and
Thus, Eq. (4.2) becomes
for all ψ i ∈ H 1(Π ϵ ).
Now let us take:
Then, Eq. (4.5) can be rewritten as follows:
where C
1 and
Let us now write Eq. (2.5) for (u
i
, v
i
) and (
Adding Eqs. (4.7) and (4.8), and applying the interpolation trace inequality (A.7) (in Appendix A) to the boundary term on Γ ϵ , one has:
where η is a small positive constant. If we choose
Summing up for i = 1, …, M and putting
where C 3 is a positive constant and, by Gronwall’s lemma, it follows that: U ≡ V ≡ 0. □
5 Homogenization
The behavior of the solutions
Lemma 5.1.
Let us consider the sets defined in Section 1.
1. There exists a linear continuous extension operator
such that
and
where C is a positive constant.
2. There exists a family of linear continuous extension operators
such that
and
where the constant C > 0 does not depend on ϵ.
Proof.
1. First we extend u into a neighbourhood X 0 of R = ∂X with smooth boundary, such that clos(X) ⊂ X 0 and clos(X 0) ⊂ int(Y). Since we have assumed that R is sufficiently smooth, we can construct a diffeomorphism as follows:
Exploiting this coordinate transformation, the function u can be extended by reflection:
Let us now consider the following smooth function
such that suppΨ ⊆ Z and Ψ = 1 in Z\X 0. Then, we define
where
Let us prove that
Taking into account the following estimates
and the Minkowski inequality, Eq. (5.15) can be rewritten as
For the derivative we obtain:
By using the Minkowski and Poincaré inequalities, Eq. (5.19) becomes:
2. The construction of
Lemma 5.2.
Let us consider the sets defined in Section 1.
1. There exists a linear continuous extension operator
such that
and
where C is a positive constant.
2. There exists a family of linear continuous extension operators
such that
and
where the constant C > 0 does not depend on ϵ.
Proof.
This Lemma can be proved by applying the same arguments considered in the proof of Lemma 5.1 and in Ref. [42] (p. 25).
Remark 5.1.
Analogous extension theorems hold also for
We briefly explain the argument for the function
If (t, x, z) ∈ [0, T] × Ω, we set
Then
so that, by Tonelli theorem
From now on, we identify
Proposition 5.1.
Let
Proof.
The convergence results (5.29)–(5.31) follow immediately from the a priori estimates given in Lemma 3.2. □
Remark 5.2.
Since
Moreover, the interpolation-trace inequality (A.7) in Appendix A and Theorem C.7 in Appendix C allow one to infer the two-scale convergence of
Next we prove the convergence of
Proposition 5.2.
Let
in the two-scale sense with
Proof.
The convergence results (5.32) follow immediately from the a priori estimates given in Lemma 3.2 and Theorem C.6 in Appendix C. □
Proof of the main Theorem 1.1.
In the case m = 1, let us rewrite Eq. (2.5) in the form:
where
where
Passing to the two-scale limit we get
An integration by parts shows that Eq. (5.35) is a variational formulation associated with the following homogenized system:
By continuity we have that
Since we have assumed that the diffusion coefficient is constant and we have proved that the limiting function v 1(t, x, z) does not depend on the microscopic variable y, Eqs. (5.36) and (5.37) reduce to:
Then,
where
with
Inserting the relation (5.44) in Eqs. (5.38) and (5.39), we get
where A is a matrix with constant coefficients defined by
Let us now rewrite Eq. (2.4) as follows:
where
where
Passing to the two-scale limit we obtain:
An integration by parts shows that Eq. (5.50) is a variational formulation associated with the following homogenized system:
To conclude, by continuity, we have that
The proof in the case 1 < m ≤ M is achieved by applying exactly the same arguments considered when m = 1.
Funding source: University of Bologna, funds for selected research topics (RFO)
Acknowledgment
S. L. is supported by GNFM of INdAM, Italy.
-
Research ethics: Not applicable.
-
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: The authors state no conflict of interest.
-
Research funding: B. F. was supported by University of Bologna, funds for selected research topics (RFO).
-
Data availability: Not applicable.
In the following, we generalize the interpolation-trace inequality given in Ref. [43]:
which is valid for any function u(x) ∈ H 1(Ω) with:
where Γ is an (n − 1)-dimensional boundary of an n-dimensional domain Ω. Indeed, in Ref. [43] (Eqs. (2.25) and (2.27), Chap. 2) the authors report an estimate similar to the one we will derive, but we will write down a short proof in order to show explicitly the dependencies of all constants from the geometry of our problem.
Let us consider now functions u(x) ∈ H 1(Ω) which do not satisfy (A.2). In this case, one can define
and apply (A.1) to (u − u 0). Hence,
where the Young inequality has been used with η being a small constant. By exploiting the Minkowski and Hölder inequalities, respectively, to estimate the terms:
Equation (A.4) reads
Theorem B.1.
(Aubin–Lions–Simon)
Let B 0 ⊂ B 1 ⊂ B 2 be three Banach spaces. We assume that the embedding of B 1 in B 2 is continuous and that the embedding of B 0 in B 1 is compact. Let p, r such that 1 ≤ p, r ≤ +∞. For T > 0, we define
(i) If p < +∞, E p,r is compactly embedded in L p ([0, T], B 1).
(ii) If p = +∞ and if r > 1, E p,r is compactly embedded in C 0([0, T], B 1).
Theorem B.2.
(Lower-semicontinuity of the norm)
Let E and F be Banach spaces and F′ be the dual space of F.
(i) Let {x n } be a sequence weakly convergent to x in E. Then, the norm on E is lower semi-continuous with respect to the weak convergence, i.e.
(ii) Let {x n } be a sequence weakly* convergent to x in F′. Then, the norm on F′ is lower semi-continuous with respect to the weak* convergence, i.e.
Definition C.1.
A sequence of functions v ϵ in L 2([0, T] × Ω) two-scale converges to v 0 ∈ L 2([0, T] × Ω × Y) if
for all
Theorem C.3.
(Compactness theorem) If v ϵ is a bounded sequence in L 2([0, T] × Ω), then there exists a function v 0(t, x, y) in L 2([0, T] × Ω × Y) such that, up to a subsequence, v ϵ two-scale converges to v 0.
Theorem C.4.
Let v ϵ be a sequence of functions in L 2([0, T] × Ω) which two-scale converges to a limit v 0 ∈ L 2([0, T] × Ω × Y). Suppose, furthermore, that
Then, for any sequence w ϵ in L 2([0, T] × Ω) that two-scale converges to a limit w 0 ∈ L 2([0, T] × Ω × Y), we have
for all
In the following, we identify H 1(Ω) = W 1,2(Ω), where the Sobolev space W 1,p (Ω) is defined by
and we denote by
Theorem C.5.
Let v
ϵ
be a bounded sequence in L
2([0, T]; H
1(Ω)) that converges weakly to a limit v(t, x) in L
2([0, T]; H
1(Ω)). Then, v
ϵ
two-scale converges to v(t, x), and there exists a function v
1(t, x, y) in
Theorem C.6.
Let v
ϵ
and ϵ∇v
ϵ
be two bounded sequences in L
2([0, T] × Ω). Then, there exists a function v
1(t, x, y) in
Theorem C.7.
Let v ϵ be a sequence in L 2([0, T] × Γ ϵ ) such that
where C is a positive constant, independent of ϵ. There exist a subsequence (still denoted by ϵ) and a two-scale limit v 0(t, x, y) ∈ L 2([0, T] × Ω; L 2(Γ)) such that v ϵ (t, x) two-scale converges to v 0(t, x, y) in the sense that
for any function
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Articles in the same Issue
- Frontmatter
- Research Articles
- Decay estimates for defocusing energy-critical Hartree equation
- A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations
- Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation
- The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy
- Regularity of center-outward distribution functions in non-convex domains
- Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities
- Periodic solutions for a coupled system of wave equations with x-dependent coefficients
- Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries
- Homogenization of Smoluchowski-type equations with transmission boundary conditions
- Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
- Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems
Articles in the same Issue
- Frontmatter
- Research Articles
- Decay estimates for defocusing energy-critical Hartree equation
- A surprising property of nonlocal operators: the deregularising effect of nonlocal elements in convolution differential equations
- Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation
- The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy
- Regularity of center-outward distribution functions in non-convex domains
- Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities
- Periodic solutions for a coupled system of wave equations with x-dependent coefficients
- Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries
- Homogenization of Smoluchowski-type equations with transmission boundary conditions
- Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
- Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems
