Abstracts
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
Matteo Bonforte
Universidad Autónoma de Madrid
We study quantitative properties of nonnegative solutions to a nonlinear and nonlocal diffusion equation of the form ut = LF (u) posed in a bounded domain, with appropriate homogeneous Dirichlet boundary conditions. The diffusion is driven by a linear operator L in a quite
general class, that includes the three most common versions of the fractional Laplacians on abounded domain with zero Dirichlet boundary conditions, as well as many other examples. The
nonlinearity F is allowed to be degenerate, the prototype being F (u) = |u|m−1 u, with m > 1.
We will shortly present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions.
Then we will concentrate on the regularity theory: space time decay and positivity, boundary
behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All
this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on
the fractional models, our techniques cover also the local case s = 1 and provide new results
even in this setting.
A surprising instance of this problem is the possible presence of non-matching powers forthe boundary behavior: this unexpected phenomenon is a completely new feature of the nonlocal
and nonlinear structure of this model, and it is not present in the semilinear elliptic case, forwhich we will shortly present the most recent results.
The above results are contained in a series of recent papers in collaboration with A. Figalli(ETH Zürich, CH), Y. Sire (J. Hopkins Univ. Baltimore, USA), X. Ros-Otón (Univ. Zürich,CH) and J. L. Vázquez (UAM Madrid, ES).

On total variation ﬂow under dynamic boundary conditions
Yoshikazu Giga
The University of Tokyo
We consider a total variation ﬂow under dynamic boundary conditions. Such a problem
appears in materials science as a prototype model. We formulate this problem as a gradient
ﬂow of a convex, lower semicontinuous energy in a Hilbert space. This enables us to constructa unique globally-in-time solution. We are also able to calculate the speed of a ﬂat portion
called a facet. However, there is a chance that boundary detachment phenomena occur. We
introduce the notion of coherency which prevents boundary detachment phenomena if the facet
is calibrable. The boundary detachment depends on mean-curvature of the boundary. It isexpected that the boundary detachment does not occur when the boundary is mean-convex. We
give several examples of radial case to support this conjecture. This is a joint work with R.
Nakayashiki (Chiba), P. Rybka (Warsaw) and K. Shirakawa (Chiba).
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Exponential stability of traveling waves fora constrained Allen-Cahn equation
Goro Akagi
Tohoku University
This talk concerns a one-dimensional Allen-Cahn equation on the whole line with thepositive-part function, which constrains the growth of each solution to be non-decreasing. Thestudy on such constrained evolution equations are motivated from phase-ﬁeld models in Damage Mechanics. Indeed, evolution of damage is supposed to be monotone due to the nature ofdamaging. In this talk, we shall discuss traveling wave dynamics, which has been well studied
for classical Allen-Cahn equations, for the constrained one. More precisely, we shall start withconstructing a one-parameter family of degenerate traveling wave solutions (identiﬁed whencoinciding up to translation) and investigate their properties. Furthermore, the traveling wave
dynamics turns out to be relevant to a free boundary problem with a peculiar motion equation
for the boundary through an analysis on a regularity issue for the constrained Allen-Cahn equation, and then, such a viewpoint enables us to prove exponential stability of degenerate travelingwaves with some basin of attraction, although they are unstable in a usual sense. This talk isbased on a joint work with C. Kuehn (München) and K.-I. Nakamura (Kanazawa).

Existence of solutions for an inhomogeneous fractional semilinear
heat equation
Kotaro Hisa
Tohoku University
In this talk, we consider necessary conditions and sufﬁcient conditions on the inhomogeneous term for the solvability of the Cauchy problem(P)∂t u + (−∆)α u = up + µ,x ∈ RN , t > 0,u(0) = 0in

RN ,where N ≥ 1, 0 < α ≤ 1, p > 1 and µ is a Radon measure or a measurable function in RN .
The existence and nonexistence of the global-in-time solution to problem (P) have been
already obtained by Bernrd [1], Zhang [3] and so on. However, there are few results on thelocal existence even in the case of α = 1. The main result of this talk is concerned with thenecessary condition for the local existence.
Theorem A ([2]). Let u be a solution of (P) in RN × (0, T ), where 0 < T < ∞. Then thereexists γ1 > 0 depending only on N , α and p such that
2pαN − p−1
(p ̸= pα ), γ1 σN[(1 )]−+1
sup µ(B(x, σ)) ≤2αT 2αx∈RN(p = pα ), γ1 log e +σ21for 0 < σ ≤ T 2α . Here pα is deﬁned bypα :=NN − 2αif 0 < 2α < Nandpα := ∞ if 2α ≥ N.

From Theorem A we observe the following: There exists C∗ > 0 such that if2pα− p−1
(p > pα ), C|x|
N[()]− 2αµ(x) ≥1−Nlog e +(p = pα ), C|x||x|(1)

in a neighborhood of the origin for some C > C∗ , then (P) ha NO local-in-time solution.
If we can prove the existence of solution when µ has the singularity as in (1), we see that(1) is the strongest singularity for the solvability of the Cauchy problem (P). In this talk,we also show the existence results. These imply that (1) is the strongest singularity of theinhomogeneous term for the solvability of the Cauchy problem (P).
References[1] G. Bernard, Existence theorems for certain elliptic and parabolic semilinear equations, J.
Math. Anal. Appl. 210 (1997), 755–776.
[2] H. Hisa, K. Ishige and J. Takahashi, Existence os solutions for an inhomogeneous fractional
semilinear heat equation, submitted, arXiv:1910.12013v1.
[3] Qi S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal.
Appl. 219 (1998), 125–139.

Singularities of solutions for the heat equation
with a dynamic Hardy potential
Eiji Yanagida
Tokyo Institute of Technology
We consider the asymptotic properties of singularities of solutions for the heat equation witha time-dependent potential. Assuming that the potential has an inverse-square leading term, weshow that there are two types of solutions with different asymptotics that are generic in somesense. For the existence, we ﬁrst ﬁnd approximate solutions and then construct suitable comparison functions by modifying singular solutions of the linear heat equation. For the classiﬁcation,
we transform the equation into an integral equation and then estimate the integral. We also show
the existence of a solution in the critical case, and study its properties. This talk is based on a
joint work with Jin Takahashi.

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Around concavity properties and heat transfer
Paolo SalaniUniversitá di Firenze
Concavity properties of solutions of PDEs are a popular subject of investigation. Amongthem, power concavities have been the most studied and log-cocavity is probably the mostfamous, due to its relevance to many applied disciplines (especially economics and statistics)
and its connection with heat transfer. A classical result by Brascamp and Lieb indeed saysthat the log-concavity of the initial datum is preserved by the heat ﬂow in a convex domain,
and this has always been considered an optimal property. Recently, in a joint work with K.
Ishige and A. Takatsu, we proved that actually a stronger concavity is preserved by the heattransfer in a natural way and we introduced a new family of concavity properties that we call”power log-concavities”. I will present some connected results and some characterizations oflog-concavity.

Elliptic and parabolic boundary value problems on rotationallysymmetric domains
Asuka Takatsu
Tokyo Metropolitan University
We study power concavity of solutions to elliptic and parabolic boundary value problems
on rotationally symmetric, strongly convex open metric balls in a Riemannian manifold. Ourresults provide a ﬁrst step to the study of power concavity for Riemannian manifolds, andimprove the known results for Euclidean spaces. This is joint work with Kazuhiro Ishige (TheUniversity of Tokyo) and Paolo Salani (University of Florence).

Blow-up of radially symmetric solutions for a semilinear heat equation
on hyperbolic space
Masahiko Shimojo
Okayama University of Science
In this talk radially symmetric solutions of a semilinear heat equation ut = ∆u + up on
the hyperbolic space are considered. First we analyze blow-up set of solutions and derive its
local blow-up proﬁle when the origin is not a blow-up point. Next the universal bounds of thenonnegative solutions are obtained together with the local blwo-up proﬁle, under the assumptionthat the exponent p is subcritical in the Sobolev sense. This is the joint work with Ai Ling Poh.

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Optimal singularities of initial functions for the solvability ofa semilinear parabolic system
Yohei Fujishima
Shizuoka University
We study the solvability for a weakly coupled system of semilinear heat equations
px ∈ RN , t > 0, ∂t u = D1 ∆u + v ,∂t v = D2 ∆v + uq ,x ∈ RN , t > 0, (u(·, 0), v(·, 0)) = (µ, ν), x ∈ RN ,(P)where D1 , D2 > 0, 0 < p ≤ q with pq > 1 and (µ, ν) is a pair of nonnegative measurable
functions in RN . We consider the existence and nonexistence of solutions for problem (P) withsingular initial functions µ and ν. In particular, we derive necessary conditions of problem (P),which enable us to identify optimal singularities of the initial functions for the solvability ofproblem (P).
This talk is based on a joint work with K. Ishige (The University of Tokyo, Japan).

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