Explore the vast range of titles available.

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. The sharp rate of convergence to positive ones was recently discussed by Bonforte and Figalli (Commun Pure Appl Math 74:744-789, 2021) based on an entropy method. An alternative proof for their result is also provided. Furthermore, the dynamics of fast diffusion flows with changing signs is discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles is proved in dumbbell domains for initial data with certain symmetry.

The present article is concerned with the Lyapunov stability of stationary solutions to the Allen–Cahn equation with a strong irreversibility constraint , which was first intensively studied in [2] and can be reduced to an evolutionary variational inequality of obstacle type. As a feature of the obstacle problem, the set of stationary solutions always includes accumulation points, and hence, it is rather delicate to determine the stability of such non-isolated equilibria. Furthermore, the strongly irreversible Allen–Cahn equation can also be regarded as a (generalized) gradient flow; however, standard techniques for gradient flows such as linearization and Łojasiewicz–Simon gradient inequalities are not available for determining the stability of stationary solutions to the strongly irreversible Allen–Cahn equation due to the non-smooth nature of the obstacle problem.

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 existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.

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 regularising terms.

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an $\\varepsilon$-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 original doubly nonlinear equation as $\\varepsilon \\to 0$. The argument relies on the suitable dualization of the former analysis of [G. Akagi and U. Stefanelli, J. Funct. Anal., 260 (2011), pp. 2541--2578] and results in a considerable extension of the possible application range of the WED functional approach to nonlinear diffusion phenomena, including the Stefan problem and the porous media equation. [PUBLICATION ABSTRACT]

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 method. The sharp rate of convergence to \\emph{positive} ones was recently discussed by Bonforte and Figalli (2021, CPAM) based on an entropy method. An alternative proof for their result will also be provided. Furthermore, dynamics of fast diffusion flows with changing signs will be discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles will be proved in dumbbell domains for initial data with certain symmetry.

The present paper is concerned with the Cauchy-Dirichlet problem for fractional (and non-fractional) nonlinear diffusion equations posed in bounded domains. Main results consist of well-posedness in an energy class with no sign restriction and convergence of such (possibly sign-changing) energy solutions to asymptotic profiles after a proper rescaling. They will be proved in a variational scheme only, without any use of semigroup theories nor classical quasilinear parabolic theories. Proofs are self-contained and performed in a totally unified fashion for both fractional and non-fractional cases as well as for both porous medium and fast diffusion cases.

This paper is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded domains, where every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile. Bonforte and Figalli (CPAM, 2021) first proved an exponential convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted \\(L^2\\) norm for smooth bounded domains by developing a nonlinear entropy method. However, the optimality of the rate remains open to question. In the present paper, their result is fully extended to possibly sign-changing asymptotic profiles as well as general bounded domains by improving an energy method along with a quantitative gradient inequality developed by the first author (ARMA, 2023). Moreover, a (quantitative) exponential stability result for least-energy asymptotic profiles follows as a corollary, and it is further employed to prove the optimality of the exponential rate.

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 in a very wide class, including the Laplacian, the \\(m\\)-Laplacian for suitable \\(m\\in(1,\\infty)\\)), the \"variable-exponent\" \\(m(x)\\)-Laplacian, or even some fractional order operators. The operator \\(A\\) is assumed to be in the form \\([A(v)](x, t) = \\alpha(x, v(x, t))\\) with \\(\\alpha\\) being measurable in \\(x\\) and maximal monotone in \\(v\\). The main results are devoted to proving existence of weak solutions for a wide class of functions \\(\\alpha\\) that extends the setting considered in previous results related to the variable exponent case where \\(\\alpha(x, v) = |v(x)|^{p(x)-2} v(x)\\). To this end, a theory of subdifferential operators will be established in Musielak-Orlicz spaces satisfying structure conditions of the so-called \\(\\Delta_2\\)-type and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators \\(A\\), \\(B\\)) to which the result can be applied.