Goro Akagi

Tohoku University | Tohokudai · Department of Mathematics

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be revealed via an energy meth...

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of...

The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of solutions as the period of oscillation goes to zero) as well as a characterization of homogenized equations....

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Fr\'emond. For the corresponding PDE system we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of r...

Constrained gradient flows are studied in fracture mechanics to describe strongly irreversible (or unidirectional) evolution of cracks. The present paper is devoted to a study on the long-time behavior of non-compact orbits of such constrained gradient flows. More precisely, traveling wave dynamics for a one-dimensional fully nonlinear Allen-Cahn t...

This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in unbounded Lipschitz domains. In particular, the forcing terms may not belong to the dual space of an energy space, e....

The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal exi...

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fra...

This paper is concerned with a parabolic evolution equation of the form A(ut)+B(u)=f$A(u_t) + B(u) = f$, settled in a smooth bounded domain of Rd$\mathbb {R}^d$, d≥1$d\ge 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, −B$-B$ stands for a diffusion operator, possibly nonlin...

This paper is concerned with a parabolic evolution equation of the form $A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, $-B$ stands for a diffusion operator, possibly nonlinear, which may range...

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy–Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles are revealed via an energy method....

This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation law, which cannot generally be realized in dissipative systems such as standard gradient flows. Main results co...

A Correction to this paper has been published: https://doi.org/10.1007/s00028-017-0390-6

This paper concerns a space-time homogenization limit of nonnegative weak solutions to porous medium equations. In particular, the so-called homogenized matrix will be characterized in terms of solutions to cell problems, which drastically vary in a scaling parameter $r > 0$. A similar problem has already been studied in [1], where the growth of th...

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of re...

This paper presents an abstract theory on well-posedness for time-fractional evolution equations governed by subdifferential operators in Hilbert spaces. The proof relies on a regularization argument based on maximal monotonicity of time-fractional differential operators as well as energy estimates based on a nonlocal chain-rule formula for subdiff...

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in...

We discuss a variational approach to abstract doubly nonlinear evolution systems defined on the time half line \(t>0\). This relies on the minimization of weighted energy-dissipation (WED) functionals, namely a family of \(\varepsilon \)-dependent functionals defined over entire trajectories. We prove WED functionals admit minimizers and that the c...

This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic o...

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In...

Let H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H . Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))} . In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}} -radially symmetric smooth functions. F...

Nondecreasing evolution is described via the coupling of an abstract doubly nonlinear differential inclusion and a constraint on the rate. The latter is formulated by imposing the monotonicity in time of the solution with respect to a given preorder in a Hilbert space. We discuss a solution notion for this problem and prove existence and long-time...

This paper presents a variational approach to doubly-nonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An elliptic-in-time regularization of the original equation ${\rm (P)}_\varepsilon$ is introduced, and then, a variational approach and a fixed-p...

The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for Fast Diffusion Equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. T...

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutio...

We present a variational approach to gradient flows of energies of the form E = phi(1) - phi(2) where phi(1), phi(2) are convex functionals on a Hilbert space. A global parameter-dependent functional over trajectories is proved to admit minimizers. These minimizers converge up to subsequences to gradient-flow trajectories as the parameter tends to...

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $\Omega$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the rela...

This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of so...

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate...

This paper is concerned with the existence of local (in time) positive solutions to the Cauchy-Neumann problem in a smooth bounded domain of RN for some fully nonlinear parabolic equation involving the positive part function r ∈ R ↦ (r)+: = r ∨ 0. To show the local solvability, the equation is reformulated as a mixed form of two different sorts of...

This paper is concerned with doubly nonlinear parabolic equations involving variable exponents. The existence of solutions is proved by developing an abstract theory on doubly nonlinear evolution equations governed by gradient operators. In contrast to constant exponent cases, two nonlinear terms have inhomogeneous growth and some difficulty may oc...

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an ε-dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the or...

This paper is concerned with stability analysis of asymptotic profiles for (possibly sign-changing) solutions vanishing in finite time of the Cauchy–Dirichlet problems for fast diffusion equations in annuli. It is proved that the unique positive radial profile is not asymptotically stable, and moreover, it is unstable for the two-dimensional annulu...

This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we also analyze the relations occurring between Lebesgue spaces of space-time variables and Lebesgue-Bochner space...

Every solution u = u(x, t) of the Cauchy–Dirichlet problem for the fast diffusion equation, ∂ t (|u|m-2u) = Δu in Ω × (0, ∞) with a smooth bounded domain Ω of ${\mathbb{R}^N}$ and 2 < m < 2* : = 2N/(N − 2)+, vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysi...

This paper is concerned with nonlinear diffusion equations driven by the p(·)-Laplacian with variable exponents in space. The well-posedness is first checked for measurable exponents by setting up a subdifferential approach. The main purposes are to investigate the large-time behavior of solutions as well as to reveal the limiting behavior of solut...

This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptotic profiles of vanishing solutions and their stability, the notions of stability and instability...

Raster images such as raster terrain maps are commonly used in computer graphics. For rapid processing such as rendering and rapid feature extraction, rapid resolution reduction methods are required that keep the quality of huge images. This study deals with the resolution reduction methods.

We discuss the existence of periodic solution for the doubly non-linear evolution equation A(u (t)) + ∂φ(u(t)) f (t) governed by a maximal monotone operator A and a subdifferential operator ∂φ in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may gen...

This paper is concerned with the initial-boundary value problem for a nonlinear parabolic equation involving the so-called p(x)-Laplacian. A subdifferential approach is employed to obtain a well-posedness result as well as to investigate large-time behaviors of solutions.

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler–Lagrange problems. Moreover, we check that m...

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: ¶V yt (u¢(t)) + ¶V j(u(t)) + B(t, u(t)) ' f(t){\partial_V \psi^t (u{^\prime}(t)) + \partial_V...

This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigrou...

This paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive Banach spaces and proving the maximal monotonicity in L p (Ω) × L p (Ω) of the nonlinear elliptic operator u → −∆ m u + β(u(·)) with a maximal monotone graph β.

The weighted energy-dissipation principle stands as a novel variational tool for the study of dissipative evolution and has already been applied to rate-independent systems and gradient flows. We provide here an example of its application to a specific yet critical doubly nonlinear equation featuring a super-quadratic dissipation.

In the present paper, we discuss the asymptotic behaviors of solutions for a couple of nonlinear parabolic equations associated with nonlinear Laplace operators and make an attempt to explain the mechanism of their behaviors by using a macroscopic random walk model.

The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u t = Δ∞ u in Ω × (0,∞), where Δ∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) W = \mathbbRN{\Omega = \mathbb{R}^N} and the initial data has a compact support; (ii) Ω is bounded and the boundary cond...

This paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive Banach spaces (cf. Brezis, Crandall and Pazy (?)) and finally proving the maximal monotonicity in Lp(Ω) ×Lp 0 (Ω) of the nonlinear elliptic operator u 7→ −∆mu + (u(·)) with a maximal monotone graph .

The existence of energy solutions to the Cauchy-Neumann problem for the porous medium equation of the form v t -Δ(|v| m-2 v)=αv with m≥2 and α∈ℝ is proved, by reducing the equation to an evolution equation involving two subdifferential operators and exploiting subdifferential calculus recently developed by the author.

The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate nonlinear parabolic equation of the form ut = D¥ uu_t = \Delta_\infty u , where D¥\Delta_\infty denotes the so-called infinity-Laplacian given by D¥ u = áD2 u Du, Du ñ\Delta_\infty u = \langle D^2 u Du, Du \rangle . To do so,...

Let V and V * be a reflexive Banach space and its dual space, respectively, and let H be a Hilbert space whose dual space H * is identified with itself H such that V → H ≡ H * → V * with continuous and densely defined canonical injections. This paper is concerned with Cauchy problems for doubly nonlinear evolution equations governed by subdifferent...

The local (in time) existence of strong solutions to Cauchy problems for doubly nonlinear abstract evolution equations with non-monotone perturbations in reflexive Banach spaces is proved under appropriate assumptions, which allow the case where solutions of the corresponding unperturbed problem may not be unique. To prove the existence, a couple o...

The existence of local (in time) solutions of the initial–boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2⩽pq+∞, Ω is a bounded domain in RN, f:Ω×(0,T)→R is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is p...

The comparison, uniqueness and existence of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate parabolic equation of the form ut = ∆∞u, where ∆∞ denotes the so-called infinity-Laplacian given by ∆∞u = N i,j=1 ux i ux j ux i x j . Our proof relies on a coercive regulariza-tion of the equation, barrier function arguments...

In this paper we report on WBT content for geography and geology using VRML. We also propose an idea of WBT content for local area study that has not yet been implemented and discuss its effect from the viewpoint of knowledge management.

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V∂φt(u(t))∋f(t), v(t)∈H∂ψ(u(t)), 0<t<T, where H∂ψ (respectively, V∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, r...

This paper is concerned with a variational inequality with a time-dependent constraint, which arises from some macroscopic models of type-II superconductivity, as well as its approximate problems associated with generalized p-Laplace operators. We prove the existence and uniqueness of solutions for each problem by establishing an abstract theory fo...

This paper is intended as an investigation of the solvability of Cauchy problem for doubly nonlinear evolution equation of the form dv(t)/dt + ∂ϕ t (u(t)) ∋ f (t), v(t) ∈ ∂ψ(u(t)), 0 < t < T , where ∂ϕ t and ∂ψ are subdifferential operators, and ∂ϕ t depends on t explicitly. Our method of proof relies on chain rules for t-dependent subdifferentials...

Asymptotic behavior of solutions of some parabolic equation associated with the p-Laplacian as p→+∞ is studied for the periodic problem as well as the initial-boundary value problem by pointing out the variational structure of the p-Laplacian, that is, ∂φp(u)=−Δpu, where φp:L2(Ω)→[0,+∞]. To this end, the no...

The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation: \(du(t)/dt + \partial \varphi (u(t)) \mathrel\backepsilon f(t),\,t \in ]0,\,T[,\) where ∂φ is the so-called subdifferential operator from a real Banach space V into its dual V*. The study in the Hilbert space...

In this paper, the asymptotic behavior of solutions for some quasilinear parabolic equation associated with p-Laplacian as p → +∞ will be discussed by in-vestigating the convergence of the functional corresponding to p-Laplacian. Moreover some abstract theory of Mosco convergence of functionals as well as evolution equations governed by subdifferen...

This paper is concerned with time-dependent constraint problems arising from macroscopic critical-state models for type-II superconductivity as well as their approximate problems associated with p-Laplacian for enough large number p. In order to derive their solvabilities, an abstract framework of doubly nonlinear evolution equations governed by ti...

Sufficient conditions for the existence of strong solutions to the Cauchy problem are given for the evolution equation du(t)/dt + partial derivative phi(1) (u(t)) - partial derivative phi(2) (u(t)) there exists f (t) in V*, where partial derivative phi(i) is the so-called subdifferential operator from a Banach space V into its dual space V (i = 1,2...

Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais gave some sufficient conditions to guarantee the so-called “Principle of Symmetric Criticality”: every critical point of J restricted on the subspace of symmetric points becomes also a critical point of J on the w...

A new framework is proposed to deal with degenerate parabolic equations such as u 1 (x,t)-Δ p |u|(x,t)- q-2 u(x,t)=f(xt),x∈Ω,t>0, where 1<p,q<+∞ and Δ p is the so-called p-Laplacian given by Δ p u:=∇·(|∇u| p-2 ∇u). Such a degenerate parabolic equation can be reduced to an abstract evolution equation governed by subdifferential operators in an appro...