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NPDE Conference

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International Conference on

Nonlinear Partial Differential Equations

A celebration of Professor Yihong Du's 60th birthday

Titles and Abstracts

Abstracts

Predator–prey models with prey-induced dispersal in a habitat with spatial heterogeneity

Inkyung Ahn (Korea University, South Korea)

Abstract: We consider a predator–prey model with nonuniform predator dispersal, called prey-induced dispersal (PYID), in which the spread of predators is small when the prey density is larger than a certain value, and when the prey density is smaller than a particular value, a large spread of predators occurs. To understand how PYID affects the dynamics and coexistence of the system in a spatially heterogeneous region, we examine a model with Holling-type II functional responses under no-flux boundary conditions wherein the predators move according to the PYID. We study the local stability of the semitrivial solution of models with PYID and linear dispersal where the predator is absent. Furthermore, we investigate the local/global bifurcation from the semitrivial solution of models with two different dispersals. If a predator’s satisfaction degree regarding the prey density is higher than a certain level, there may exist a case that is not beneficial for predators in terms of their fitness.

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Equilibria of vortex type Hamiltonians on closed surfaces and point vortex dynamics

Thomas Bartsch (Univ. Giessen, Germany)

Abstract: The Hamiltonian system describing the dynamics of \(N\) point vortices on a surface appears as a singular limit for the dynamics of vortices in the Euler equations for an ideal fluid, or for Gross-Pitaevskii dynamics. Equilibria also yield blow-up families of solutions of mean field equations near critical parameter values. We present recent results on the existence of equilibria and on the point vortex dynamical system on a closed surface.

The talk is based on joint work with Mohameden Ahmedou and Tim Fiernkranz.

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Isoperimetric sets for weighted twisted eigenvalues

Barbara Brandolini (Università degli Studi di Palermo, Italy)

Abstract: In this talk we discuss a shape optimization problem for the first twisted eigenvalue of the weighted operator \(L=-\textrm {div}\left (\gamma (x)\nabla \right )\) among sets with fixed weighted measure. If \(\gamma (x)\) satisfies some convenient assumptions, like in the cases \(\gamma (x)=e^{-|x|^2}\) or \(\gamma (x)=x_N^k \> (k \ge 0)\), we show that this eigenvalue is minimized by the union of two disjoint isoperimetric sets sharing the same measure.

The talk is based on a recent work in collaboration with Antoine Henrot, Anna Mercaldo and Maria Rosaria Posteraro.

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Asymptotic behaviour of problems of the (p,q)-Laplacian types

Michel Chipot (Institute of Mathematics, University of Zurich, Switzerland)

Abstract: The goal of this paper is to study the asymptotic behaviour of a linear combination of operators of the p-Laplacian type in cylinder like domains having some directions going to infinity. The case of a combination of two such operators has been the topic of numerous studies in the recent years.

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Bistable pulsating fronts in slowly oscillating environments

Weiwei Ding (South China Normal University, China)

Abstract: In this talk, I will present some recent progress on reaction-diffusion fronts in spatially periodic bistable media. The results include: existence of pulsating fronts with large periods, existence of and an explicit formula for the limit of front speeds as the spatial period goes to infinity, convergence of pulsating front profiles to a family of front profiles associated with spatially homogeneous equations.

This talk is mainly based on joint work with Francois Hamel and Xing Liang.

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The strange behaviour of nonlocal minimal surfaces

Serena Dipierro (University of Western Australia, Australia)

Abstract: Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors than the ones minimizing the classical perimeter. Among these peculiar features, an interesting property, which is also in contrast with the pattern produced by the solutions of linear equations, is given by the capacity, and the strong tendency, of adhering at the boundary.

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Degenerate parabolic PDEs and their attractors

Messoud Efendiev (Helmholtz Center Munich/TUM, Germany)

Abstract: In my talk I consider a new class of degenerate parabolic PDEs arising in the modeling of life sciences. The long-time dynamics of solutions is studied in terms of their attractors. Some open problems will also be discussed.

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On the number of critical points of solutions of elliptic equations

Massimo Grossi (Sapienza Università di Roma, Italy)

Abstract: The calculation of the number of critical points of a solution of a PDE is an old and classic problem. Some powerful techniques (Morse theory, topological degree) allow to give estimates on the total number of critical points. However, the exact calculation requires additional ideas. In the famous Gidas-Ni-Nirenberg Theorem (see [1]) the uniqueness of the critical point of the solution to \begin{equation*} \begin{cases} -\Delta u =f(u)&in\ \Omega \\ u>0&in\ \Omega \\ u=0&on\ \partial \Omega \end{cases} \end{equation*} in convex and symmetric domains was proved. If we drop the symmetry assumption then the problem because much more difficult and in this case the uniqueness of the critical point was proved only for special \(f\) (torsion problem, first eigenfunction).

In this lecture we consider contractible domains allowing that the curvature of \(\partial \Omega \) could be negative.

Finally the case of the second eigenfunction in domains with large eccentricity will be considered.

This is joint work with Fabio De Regibus (University of Roma “La Sapienza’) and Francesca Gladiali (University of Sassari, Italy).

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Liouville-type Theorems for steady solutions to the Navier-Stokes system in a slab

Changfeng Gui (University of Texas at San Antonio, United States)

Abstract: In this talk, I will present recent results on Liouville-type theorems for the steady incompressible Navier-Stokes system in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or \(ru^r\) is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl velocity is independent of the angular variable, or \(ru^r\) decays to zero as \(r\) tends to infinity, The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions.

The talk is based on recent joint work with Jeaheang Bang, Yun Wang and Chunjing Xie.

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Non-degeneracy of the blowing-up solution for Lane Embden

Yuxia Guo (Tsinghua University in Beijing, China)

Abstract: In this talk, we consider the following elliptic system \begin{equation} \begin{cases} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), &\hbox { in }\Omega , \\-\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &\hbox { in }\Omega , \\u=v=0&\hbox { on }\partial \Omega , \end{cases} \end{equation} where \(\Omega \) a smooth bounded domain in \(\mathbb {R}^{N}\), \(N\geq 3\), \(\epsilon \) is a small parameter, \(\alpha \), \(\beta _1\) and \(\beta _2\) are real numbers, and \((p,q)\) is a pair of positive numbers lying on the critical hyperbola \begin{equation} \begin{split} \frac {1}{p+1}+\frac {1}{q+1} =\frac {N-2}{N}. \end{split} \end{equation} We first revisited the multiple blowing-up solutions constructed in an earlier paper and then we prove its non-degeneracy via the local Pohozaev identities. We believe that the variety new ideas and technique computations we used here would be very useful to deal with other related problems involving Lane-Emden system and critical exponents.

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Singular points and singular solutions for some elliptic equations

Zongming Guo (Henan Normal University, China)

Abstract: We are interested in asymptotic behaviors at isolated singular points of solutions for some elliptic equations. Optimal regularity of solutions for some elliptic equations with singular weights is determined by asymptotic expansions at the singular points of prescribed solutions. We can also construct singular solutions to the equations via asymptotic behaviors at the singular points and the fixed point argument.

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Convergence to a self-similar solution for a one-phase Stefan problem arising in corrosion theory

Danielle Hilhorst (CNRS and University Paris-Saclay, France)

Abstract: Steel corrosion plays a central role in different technological fields. In this talk, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. We prove that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as time tends to infinity. We do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations. In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

This is joint work with Meriem Bouguezzi, Yasuhito Miyamoto and Jean-François Scheid.

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On the \(\sigma _k\)-Loewner–Nirenberg problem

Yanyan Li (Rutgers University, United States)

Abstract: We study the regularity of the viscosity solution u of the \(\sigma _k\)-Loewner–Nirenberg problem on a bounded smooth domain \(\Omega \subset \Omega \subset \Bbb R^n\) for \(k\ge 2\). It was known that u is locally Lipschitz in \(\Omega \). We prove that, with \(d\) being the distance function to \(\partial \Omega \) and \(\delta >0\) sufficiently small, \(u\) is smooth in. \(\{ 0<d(x)<\delta \}\) and the first \((n-1)\) derivatives of \(d u^{2/(n-2)} \) are Hölder continuous in \(\{0\le d(x)<\delta \}\). Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when \(\partial \Omega \) contains more than one connected components, \(u\) is not differentiable in \(\Omega \).

This is joint work with Luc Nguyen and Jingang Xiong.

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Some applications of optimal transportation

Jiakun Liu (University of Wollongong, Australia)

Abstract: In this talk, we will introduce some interesting applications of optimal transportation in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric inequality in geometry; and an application in image recognition relating to a transport between hypercubes.

This talk is based on joint work with Shibing Chen, Xu-Jia Wang and Gregoire Loeper.

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General convergence results for bounded solutions of nonlinear diffusion equations

Bendong Lou (Shanghai Normal University, China)

Abstract: I will talk about the general convergence results for bounded solutions of several kinds of one dimensional parabolic problems, including the Cauchy problem of reaction diffusion equations (RDEs), RDEs with Stefan boundary conditions and the Cauchy problem of porous media equations. We also give the application of these general results on the asymptotic behavior for the solutions of the bistable diffusion equations.

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Stability of fronts in bidomain models

Hiroshi Matano (Meiji University, Japan)

Abstract: Bidomain models are widely used to simulate electrical signal transmissions in the heart. Mathematically bidomain models are expressed in terms of pseudo-differential equations involving a Fourier integral operator that is anisotropic. Despite their importance in cardiac electrophysiology, systematic mathematical studies of bidomain models started only relatively recently. As the bidomain models usually have strong anistropy, the stability of fronts depends on the direction of its motion. In 2016, we considered bidomain Allen-Cahn type equations on \({\mathbb R}^2\) and revealed the deep relation between the linear stability of a planar wave and the so-called Frank diagram (joint work with Yoichiro Mori, CPAM 2016). In this talk, I will present our recent results on the nonlinear stability of a planar wave of bidomain Allen-Cahn type equations and also show some numerical simulations of pulse waves of the bidomain FitzHugh-Nagumo system that are noticeably different from those of the conventional FitzHugh-Nagumo systems.

This is joint work with Yoichiro Mori, Mitsunori Nara and Koya Sakakibara.

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Nonlinear Stefan problem with a certain class of multi-stable nonlinearity

Hiroshi Matsuzawa (Kanagawa University, Japan)

Abstract: In this talk I will discuss the long-time behavior of solutions to the following nonlinear Stefan problem \begin{equation*} \left \{\begin{array}{ll} u_t=\Delta u+f(u), & t>0,\ x\in \Omega (t), \\ u=0,\ u_t=\mu |\nabla _x u|^2, & t>0,\ x\in \Gamma (t)=\partial \Omega (t), \\ u(0, x)=u_0(x), & x\in \Omega _0=\Omega (0) \end{array}\right . \end{equation*} with a certain class of multi-stable nonlinearity \(f\). I will present that when \(f\) is a positive bistable type nonlinearity, the long-time behavior of the solutions are classified into four cases: vanishing, small spreading, big spreading and transition. In particular, I will report that when transition occurs for the solution \(u\), there exists a point \(x_0\in \mathbb {R}^N\), \(u(t,\,\cdot \,)\) converges an equilibrium solution with radially symmetric and radially decreasing about some \(x_0\) as \(t\to \infty \) locally uniformly on \(\mathbb {R}^N\).

This talk is based on joint work with Dr. Yuki Kaneko (Japan Women’s University) and Professor Yoshio Yamada (Waseda University).

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Boundary Layer Solutions of a Singularly Perturbed Gierer-Meinhardt System

Linfeng Mei (Zhejiang Normal University, China)

Abstract: We consider boundary layer solutions of a singularly perturbed Gierer-Meinhardt system. The existence and profile, as well as the linear stability of boundary layer solutions were studied. When the boundary layer solutions is unstable, spike solutions near the boundary layer were also studied.

This is joint work with Prof. Juncheng Wei and Dr. Daniel Gomez of UBC.

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Eventual positivity and the bi-Laplacian

Jonathan Mui (University of Sydney, Australia)

Abstract: I will give a brief survey of recent results on eventually positive semigroups of operators and the connection with concrete PDE problems involving the bi-Laplace operator.

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Leapfrogging vortex rings for the 3 dimensional incompressible Euler equations

Monica Musso (University of Bath, United Kingdom)

Abstract: A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid 3-dimensional fluid. Helmholtz (1858) observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. In this talk I will present a mathematical justification for this phenomenon by constructing a smooth solution of the 3d Euler equations exhibiting this motion pattern. This work is in collaboration with J Davila, M. del Pino and J. Wei.

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Regularity properties in a class of nonlinear partial differential equations

Antonella Nastasi (University of Palermo, Italy)

Abstract: We study quasiminimizers, that minimize the energy functional up to a multiplicative constant, of the following anisotropic energy \((p,q)\)-Laplace Dirichlet integral \begin{equation*} \int _{\Omega } a g_{u}^p d \mu + \int _{\Omega }b g_{u}^q d \mu , \end{equation*} in metric measure spaces, with \(g_u\) the minimal \(q\)-weak upper gradient of \(u\). Here, \(\Omega \subset X\) is an open bounded set, where \((X, d, \mu )\) is a complete metric measure space with metric \(d\) and a doubling Borel regular measure \(\mu \), supporting a weak \((1,p)\)-Poincaré inequality for \(1<p<q\). We consider some coefficient functions \(a\) and \(b\) to be measurable and satisfying \(0<\alpha \leq a,b\leq \beta \), for some positive constants \(\alpha \), \(\beta \). Recently, many authors have focused their attention to \(p\), \(p(x)\), \((p,q)\) and fractional Laplace operators and to the calculus of variations issues associated to them (see [1], [5], [2]).

Using a variational approach, we study interior regularity for quasiminimizers of the above integral, as well as regularity results up to the boundary. Indeed, variational methods are powerful tools not only for the existence of solutions, but also for obtaining information on the behavior and regularity properties of minimizers and, more generally, quasiminimizers. We note that minimizers are solutions to the corresponding Euler-Lagrange equation, but quasiminimizers are not directly related to any partial differential equation. Thus the only possibility is to develop variational methods that are related to the energy functional (see [4], [3]). More specifically, for the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider \((p,q)\)-minimizers and we give an estimate for their oscillation at boundary points.

This is joint work [4] with Cintia Pacchiano Camacho (Aalto University).

References

[1]   M. Chipot, H.B. de Oliveira, Some results on the \(p(u)\)-Laplacian problem. Math. Ann. 375 (2019), \(n^o\) 1-2, 283–306.

[2]   M. Colombo, G. Mingione, Regularity for double phase variational problems Arch. Ration. Mech. Anal., 215 (2015), 443–496.

[3]   P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations, 90 (1991), 1–30.

[4]   A. Nastasi, C. Pacchiano Camacho, Regularity properties for quasiminimizers of a \((p,q)\)-Dirichlet integral, Calc. Var., 227 (60) (2021).

[5]   Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 1842–1864.

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Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with large advection and general boundary conditions

Rui Peng (Zhejiang Normal University, China)

Abstract: In this talk, we shall report our recent progress on the principal eigenvalue of a linear second order elliptic operator complemented by the Dirichlet boundary condition or a general Robin boundary condition. In the higher dimensional case, under proper conditions on the advection function, we establish the asymptotic behavior of the principal eigenvalue as the advection coefficient converges to infinity, and in one dimensional case, we obtain a complete description for such asymptotic behavior provided the derivative of the advection function changes sign at most finitely many times. Our work improves or complement some exsiting results by Chen-Lou and P.-Zhou, and answer some open questions raised by Berestycki-Hamel-Nadirashvili.

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Elliptic systems with critical growth

Angela Pistoia (Sapienza Università di Roma, Italy)

Abstract: I will present some old and new results concerning existence of sign-changing solutions to the Yamabe problem on the round sphere and existence of positive solutions to a class of systems of PDE’s with critical growth in the whole euclidean space in presence of a competitive regime.

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Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

Fernando Quirós (Universidad Autónoma de Madrid, Spain)

Abstract: We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution with this mass of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We assume in principle that the convection nonlinearity is locally Lipschitz, but we are also able to cover some situations in which it is only locally Hölder. This possibility has never been considered before in the nonlocal diffusion setting.

This is joint work with Jørgen Endal (Norwegian University of Science and Technology) and Liviu Ignat (Simion Stoilow Institute of Mathematics of the Romanian Academy).

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Optimal pointwise control for high order nonlinear equations

Frédéric Robert (Université de Lorraine, France)

Abstract: In this talk, I will discuss the non-compactness properties of families of solutions to nonlinear equations such as \[P u= u^{\frac {n+2k}{n-2k}}\tag {E}\] on a manifold \((M^n,g)\) where \(P=(-\Delta _g)^k+\dots \) is an elliptic operator of order \(2k\). Examples of such operators are the conformal Laplacian (\(k=1\)), the Paneitz operator (\(k=2\)) and the GJMS operators (\(k>2\)). The corresponding nonlinear equation is the the \(Q\)-curvature equation.

Families of solutions \((u_i)_i\) to \((E)\) might blow-up and develop a singularity, called a peak, which is explicit. We prove a sharp pointwise control by the peak, namely: there exists \(C>0\) such that \[|u_i(x)|\leq C\cdot \text {peak}.\] The main difficulty is that one cannot use the classical method for \(k=1\), which relies on the comparison principle that does not hold for \(k>1\).

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Influence of Road-field Diffusion on the Invasion and Competition of Aedes Albopictus and Aedes Aegypti Mosquitoes

Shigui Ruan (University of Miami, United States)

Abstract: Based on the invasion of Aedes albopictus mosquito along roads and its competition with Ae. aegypti mosquito in Florida, we propose a two-species competition model with road-field diffusion in which the invasive population not only disperses in the interior of the spatial domain but also moves faster on the boundary of the domain. In the case of strong-weak competition where the invasive species is stronger than the local one, it is shown that solutions converge uniformly to a semi-positive equilibrium such that the invasive species drives the local species out of its habitat. In the case of weak-weak competition, solutions converge uniformly to a positive equilibrium such that both species coexist. Moreover, it is shown that the asymptotic spreading speed of the wave fronts is increasing only if the road diffusion rate is greater than the field diffusion rate. Otherwise, the asymptotic spreading speed of the wave fronts does not change. Numerical simulations are presented to explain the current estimated distributions of these two mosquito species in Florida.

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Population dynamics under climate change: persistence criterion and effects of fluctuations

Wenxian Shen (Auburn University, United States)

Abstract: The current talk is concerned with population dynamics under climate change. The evolution of species is modelled by a reaction-diffusion equation in a spatio-temporally heterogeneous environment described by a climate envelope that shifts with a time-dependent speed function. For a general almost-periodic speed function, we establish the persistence criterion in terms of the sign of the approximate top Lyapunov exponent and, in the case of persistence, prove the existence of a unique forced wave solution that dominates the population profile of species in the long run. In the setting for studying the effects of fluctuations in the shifting speed or location of the climate envelope, we show by means of matched asymptotic expansions and numerical simulations that the approximate top Lyapunov exponent is a decreasing function with respect to the amplitude of fluctuations, yielding that fluctuations in the shifting speed or location have negative impacts on the persistence of species, and moreover, the larger the fluctuation is, the more adverse the effect is on the species. In addition, we assert that large fluctuations can always drive a species to extinction.

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Phytoplankton competition for nutrients and light in a stratified lake: a mathematical model connecting epilimnion and hypolimnion

Junping Shi (College of William & Mary, United States)

Abstract: Various mathematical models describing vertical distribution of phytoplankton in the water column will be introduced. In particular, we will introduce a new mathematical model connecting epilimnion and hypolimnion to describe the growth of phytoplankton limited by nutrients and light in a stratified lake. Stratification separates the lake with a horizontal plane called thermocline into two zones : epilimnion and hypolimnion. The epilimnion is the upper zone which is warm (lighter) and well-mixed; and the hypolimnion is the bottom colder zone which is usually dark and relatively undisturbed. The growth of phytoplankton in the water column depends on two essential resources: nutrients and light. The critical thresholds for settling speed of phytoplankton cells in the thermocline and the loss rate of phytoplankton are established, which determine the survival or extirpation of phytoplankton in epilimnion and hypolimnion.

This is joint work with Jimin Zhang (Heilongjiang University), Jude Kong (York University) and Hao Wang (University of Alberta).

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Proving Harnack inequalities via a multipoint maximum principle approach

Jessica Slegers (University of Sydney, Australia)

Abstract: In this talk, we present a new method of proving global pointwise Harnack inequalities for positive solutions of parabolic equations, such as the classical heat equation, porous medium equation, and \(p\)-heat equation. Our approach is based on a multipoint maximum principle argument, which does not rely on additional estimates such as those by Aronson-Benilan or Esteban-Vazquez. We demonstrate our main techniques by providing a new proof of Li-Yau’s celebrated Harnack inequality for positive solutions of the heat equation on \(\mathbb {R}^d\). This talk is based on joint work with Ben Andrews and Daniel Hauer.

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Rotating Spirals in segregated reaction-diffusion systems

Susanna Terracini (University of Turin, Italy)

Abstract: We give a complete characterization of the boundary traces \(\varphi _i\) (\(i=1,\dots ,K\)) supporting spiraling waves, rotating with a given angular speed \(\omega \), which appear as singular limits of competition-diffusion systems of the type \begin{equation*} \begin{cases} \partial _t u_i -\Delta u_i = \mu u_i -\beta u_i \sum _{j \neq i} a_{ij} u_j &\text {in } \Omega \times \mathbb R^+ \\ u_i = \varphi _i &\text {on $\partial \Omega \times \mathbb R^+$}\\ u_i(\boldsymbol x,0) = u_{i,0}(\boldsymbol x) &\text {for $ \boldsymbol x \in \Omega $} \end{cases} \end{equation*} as \(\beta \to +\infty \). Here \(\Omega \) is a rotationally invariant planar set and \(a_{ij}>0\) for every \(i\) and \(j\). We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by \(\omega \in \mathbb R\), which reduce to homogeneous harmonic polynomials for \(\omega =0\).

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Mean First Passage Time Problems

William Trad (University of Sydney, Australia)

Abstract: Recent results on mean first passage time problems within Riemannian 3-manifolds as domains will be described in conjunction with the microlocal analytic methods used to derive the boundary restricted Greens function for a mixed boundary value problem. In addition, these methods have extended to the derivation of eigenvalue asymptotics for the Laplacian with Neumann boundary conditions on Riemannian 3-manifolds.

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The Pucci conjecture

Neil Trudinger (Australian National University, Australia)

Abstract: The Pucci conjecture concerns a sharp form of the maximum principle for linear second order, uniformly elliptic partial differential equations in therms of an integral norm of the inhomogeneous term and an ellipticity constant. We report here on recent improvements to its partial solution in previous work.

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Long-range phase coexistence models

Enrico Valdinoci (University of Western Australia, Australia)

Abstract: We will discuss classical and recent results concerning the Allen-Cahn equation and its long-range counterpart, especially in relation to its limit interfaces, which are (possibly nonlocal) minimal surfaces, and to the corresponding rigidity and symmetry properties of flat solutions.

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Positive solutions of a diffusive eco-epidemiological model with Dirichlet boundary conditions

Mingxin Wang (Henan Polytechnic University, China)

Abstract: In this talk we study positive solutions of a diffusive eco-epidemiological model with Dirichlet boundary conditions. The existence and uniqueness of positive solutions are obtained. Moreover, the existence, uniqueness, non-degeneracy and globally asymptotically stablities of semi-trivial nonnegative solutions are also attained.

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The Interaction of Traveling Waves with Transition Layers for a SKT Competition Model with Cross-Diffusion

Yaping Wu (Capital Normal University, China)

Abstract: Consider the following SKT competition model \[ \left \{ \begin{array}{ll} u_t=\epsilon ^2 u_{xx}+(a_1-b_1u-c_1 v)u,\\ v_t=[(1+\gamma u)v]_{xx}+(a_2-b_2u-c_2v)v, \end{array} x\in \mathbb R,\; t>0; \right . \] with small diffusion rate \(\epsilon >0\) and non-small cross diffusion rate \(\gamma >0\), \(a_i,b_i,c_i>0\)(\(i=1,2\)) and \(d_2>0\). By applying geometric singular perturbation argument and topological index method, Yaping Wu and Ye Zhao proved that the system in strong competition case has a family of stable travelling waves connecting \((0,a_2/c_2)\) and\((a_2/b_2,0)\) with locally unique slow speed \(c=\epsilon c_\epsilon \) and both components of the waves have transition layers. In this talk we shall mainly talk about our recent work on the higher order two-scale expansion of the waves with transition layer for small \(\epsilon >0\) and the long time behavior and the asymptotic speed of the shifts of two interacting waves.

It is joint work with Dr. Hao Zhang (Capital Normal University) and Prof. Shin Ichiro Ei (Hokkaido University).

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Periodic Logistic Solutions on Domains with Refuges

Zeaiter Zeaiter (University of Sydney, Australia)

Abstract: We consider the parameter dependent periodic-logistic problem with a logistic degeneracy that replicates refuges in the habitat of a population. \begin{equation*} \begin{aligned} \frac {\partial u}{\partial t} + \mathcal {A}(t)u &= \mu u - b(x,t)g(x,t,u)u& &\text {in }\Omega \times (0,T)\\ \mathcal {B}(t)u &= 0& &\text {on }\partial \Omega \times [0,T]\\ u(\cdot ,0) &= u(\cdot ,T)& &\text {in }\Omega \end{aligned} \end{equation*} Working under no assumptions of the boundary regularity of the domain we show the existence of a periodic solution which bifurcates with respect to the parameter. Moreover, we give sufficient conditions for the case where the periodic solution is stable in time with respect to solutions to the equivalent initial-boundary value logistic problem.

This is joint work with Daniel Daners

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