Special Session 173: Mathematical and Numerical Analysis on Nonlinear PDEs

Lifespan estimates for the semilinear wave equation with a forcing term

Meiirkhan B Borikhanov Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):    H.Takamura, B.T. Torebek

Abstract: We study the lifespan of solutions to a semilinear wave equation with a forcing term. Under suitable assumptions on the initial data and the external force, we derive upper and lower bounds for the maximal existence time of solutions. The analysis highlights how the forcing term influences blow-up behavior and modifies the classical lifespan estimates known for the unforced problem.

Lp-Lq estimates for evolution equations with damped oscillations

Marcello DAbbicco University of Bari Italy Co-Author(s):    Marcelo Rempel Ebert

Abstract: In this talk, we discuss $L^p-L^q$ estimates, with $1\leq p\leq q\leq\infty$, for dissipative wave-type equations, under the assumption that the dissipation dampens the oscillations but it does not cancel them. We employ methods based on Fourier analysis, including the use of real Hardy space $H^1(\mathbb{R}^n)$ and stationary phase arguments to take into account of the dispersive nature of the equation.

Scattering and energy cascade for the 2D Klein-Gordon-Zakharov

Shijie Dong Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Zihua Guo and Kuijie Li

Abstract: We study the Klein-Gordon--Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp long-time behavior, including sharp time decay, scattering, and growth of Sobolev norms. This is joint with Prof. Zihua Guo and Prof. Kuijie Li.

Global existence for a semilinear damped wave equation with certain positive initial displacement and negative initial velocity

Kazumasa Fujiwara Ryukoku University Japan Co-Author(s):    Vladimir Georgiev

Abstract: In this talk, global existence for the Cauchy problem for a semilinear damped wave equation with positive initial displacement and negative initial velocity is investigated. In the classical theory, solutions are known to blow up if the sum of the mean values of the initial displacement and velocity is nonnegative, while global existence holds if the initial data satisfy a pointwise sign condition. However, it remains unclear whether solutions blow up in other cases. In this talk, a new condition on the initial data ensuring global existence is presented. Our approach is to show that the solution satisfies the classical global existence condition of Li and Zhou at some time. In particular, the solution is bounded by the initial displacement times a time-dependent amplification factor. This factor is shown to change sign at a certain time. The initial displacement and velocity are assumed to be given by a common polynomially decaying function multiplied by a positive constant and a negative constant, respectively.

On lifespan estimates for a discrete Fujita equation

Kohei Higashi Musashino University Japan Co-Author(s):

Abstract: We study the lifespan of solutions to a discrete analogue of the semilinear heat equation with power nonlinearity, viewed as a discrete Fujita equation on the lattice. Previous work has shown that this model exhibits finite-time blow-up with the same Fujita critical exponent as in the continuous setting. We investigate how the lifespan depends on the size of small initial data and on the exponent relative to the Fujita critical exponent. We prove that, in both the subcritical and critical cases, the lifespan has the same order as that of the corresponding continuous problem.

On inverse scattering for time-decaying harmonic and repulsive potentials

Atsuhide Ishida Tokyo University of Science Japan Co-Author(s):

Abstract: We study quantum inverse scattering for Schr\"{o}dinger equations with time-dependent quadratic Hamiltonians.The unperturbed system contains either harmonic or repulsive quadratic potentials whose strength decays at a critical rate in time. This decay produces scattering states that differ from both the free and time-independent cases. For real-valued interaction potentials satisfying suitable spatial decay conditions, the existence of wave operators has been established in previous work, and the associated scattering operator is well defined. The main result presented here proves uniqueness in inverse scattering: the interaction potential is uniquely determined by the scattering operator. In the harmonic case, scattering arises despite the confining nature of the quadratic potential. In the repulsive case, the time decay modifies the long-time behavior and allows decay comparable to the long-range class in the Stark effect, while remaining short-range within this framework.

The lifespan of solutions of semilinear wave equations with characteristic weights in two space dimensions

Masakazu Kato Graduate School of Science, University of Hyogo Japan Co-Author(s):    Hiroyuki Takamura, Kyouhei Wakasa

Abstract: In this talk, we study the initial value problem for semilinear wave equations with characteristic weights in two space dimensions. Our purpose of this talk is to determine the critical exponent that separates the time global existence and non-existence of solutions and the sharp lifespan estimates. We classify the exponent and lifespan by the integral of higher-order moments of the initial positions and speeds.

Space-time decay of global solutions to a system of damped wave equations

Kosuke Kita Hokkaido university Japan Co-Author(s):

Abstract: We consider the Cauchy problem for a weakly coupled system of semilinear damped wave equations in \(\mathbb{R}^N\) (\(N=1,2,3\)). We prove the global existence of mild solutions for sufficiently small initial data and derive weighted pointwise space-time decay estimates. These estimates provide a refined description of the decay of each component in time and space. In particular, depending on the nonlinearity exponent relative to the Fujita critical exponent, the solutions exhibit different decay patterns, including an asymmetric regime in which one component decays more slowly and a logarithmic correction in the borderline case. The proof is based on weighted \(L^\infty\)-estimates for the linear damped wave equation.

Global Weak Solutions to Nonlinear Wave Equations with Damping under Fractional Derivative Control

Hideo Kubo Hokkaido University Japan Co-Author(s):    Bushra Nisar

Abstract: In this talk we present a result on the initial-boundary value problem for the wave equation with nonlinear damping and source terms prescribed the nonhomogeneous Neumann boundary condition involved time fractional derivative of solution. In the pioneering work of Professors Georgiev and Todorova (1994), existence of global weak solution exists for homogeneous Dirichret boundary value problem, provided the damping term is dominated to the source term in some sense. On the other hand, a recent work of Mbodje(2004) consider the linear wave equation with the time fractional Neumann boundary condition in one space dimension. Out purpose is simply to extend the latter result to the nonlinear setting as in the former work.

Asymptotic expansions of global solutions to the convection-diffusion equation with critical dissipation

Ryunosuke Kusaba Tohoku University Japan Co-Author(s):    Taiki Takeuchi

Abstract: This talk is concerned with the large-time behavior of global solutions to the Cauchy problem for the convection-diffusion equation with critical dissipation. The existence, decay estimates, and the first and second order asymptotic expansions of global solutions are established by using maximal regularity estimates in the homogeneous Besov spaces. In addition, with the aid of the self-similar structures in the asymptotic profiles, the optimal decay rates of the global solutions and the optimal convergence rates for the first order asymptotic expansion are determined.

Wave type equations with perturbed derivatives

Sandra Lucente Dipartimento Interuniversitario di Fisica, Bari University Italy Co-Author(s):    Felisia Chiarello, Giovanni Girardi

Abstract: We investigate semilinear wave-type equations that can be recast as wave equations with derivatives perturbed by zero-order terms. This framework covers several well-studied cases, including the scale-invariant wave equation. In this setting, we refine existing blow-up results for radial initial data with suitable decay, and identify conditions on the zero-order terms that govern the interplay between derivative perturbations, initial data size, and nonlinearity exponent.

Analytical and numerical results on the global bifurcation structure of a cell polarization problem

Tatsuki Mori Musashino University Japan Co-Author(s):    Tatsuki Mori

Abstract: We investigate the global bifurcation structure of solutions for a nonlinear boundary value problem with a nonlocal constraint arising in a cell polarization model with mass conservation proposed by Mori, Jilkine, and Edelstein-Keshet. Regarding the shadow system of the stationary problem, a representation formula for the solutions and the global bifurcation diagrams have already been obtained. In this talk, we derive a representation formula for the solutions of the original problem and report on numerical simulations of its global bifurcation diagram. In particular, global bifurcation diagrams include secondary and imperfect bifurcation phenomena.

Blowing-up solutions of Klein-Gordon equations with gauge variant semilinear terms in FLRW spacetimes

Makoto Nakamura The University of Osaka Japan Co-Author(s):    Makoto Nakamura, Takuma Yoshizumi

Abstract: Blowing-up solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lema\^itre-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied through the scale-function and the curved mass.

Blow-up results for Nakao-type problems with time-dependent coefficients

Alessandro Palmieri University of Bari Italy Co-Author(s):    Yuequn Li

Abstract: In this talk, I will discuss some blow-up results for a Nakao-type semilinear weakly coupled system. In particular, we consider the case when a wave equation is coupled with a damped wave equation with a time-dependent coefficient for the damping term. We investigate the influence of a scattering producing/scale-invariant damping term on the blow-up region for power nonlinearities and for derivative-type nonlinear terms.

Numerical quenching time for rescaling algorithm on nonlinear wave equations

Takiko Sasaki Musashino University Japan Co-Author(s):    Maiss\hat{a} Boughrara

Abstract: The purpose of this talk is to establish numerical methods for computing quenching solutions of one space dimensional nonlinear wave equations with power nonlinearities. We consider a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. To numerically reproduce the quenching phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. After having verified the convergence of our proposed schemes, we prove that solutions of those finite-difference schemes actually quench in the corresponding quenching times. Then, we prove that the numerical quenching time converges to the exact quenching time as the discretization parameters tend to zero. Several numerical examples that confirm the validity of our theoretical results are also offered.

Fractional dispersion phenomena in the Helmholtz equation

Nico Michele Schiavone Polytechnic University of Madrid Spain Co-Author(s):

Abstract: We discuss the fractional dispersion of solutions to the Helmholtz equation with periodic scattering data. Under suitable rescaling, the interaction between different frequencies exhibits the same fluctuating behavior found in the Schr\{o}dinger equation. Since the Helmholtz equation represents the stationary form of various evolution equations, these phenomena appear to be a general feature of periodic scattering at the fractional scale. Our results are based on establishing an asymptotic fractional uncertainty principle for solutions to the Helmholtz equation. Joint work with Javier Canto (UPV/EHU) and Luis Vega (UPV/EHU, BCAM).

Discontinuous Galerkin Methods for the Incompressible Magnetohydrodynamic System with Navier-Type Boundary Conditions

Nour Seloula University of Caen Normandie France Co-Author(s):

Abstract: In this talk, we introduce and analyze a discontinuous Galerkin method for the stationary incompressible magnetohydrodynamic system subject to Navier-type boundary conditions for both the velocity and the magnetic field. We establish a new discrete Sobolev inequality in the $L^p$ setting, which plays a central role in the analysis, in particular for proving the well-posedness and convergence of the scheme. The existence and uniqueness of a discrete solution are obtained by means of the Banach fixed point theorem under a smallness assumption on the data. Furthermore, we derive a priori error estimates in a natural energy norm for both the velocity and the magnetic field. To the best of our knowledge, this is the first work providing a complete analysis of a discontinuous Galerkin method for the nonlinear coupled magnetohydrodynamic system with Navier-type boundary conditions imposed simultaneously on the velocity and the magnetic field.

Quenching of finite-difference solutions to nonlinear heat equations with non-local terms

Tetsuji Tokihiro Musashino University Japan Co-Author(s):    Takiko Sasaki

Abstract: This talk investigates the quenching phenomenon for numerical solutions of nonlinear heat equations incorporating non-local terms. Quenching occurs when a solution remains bounded while its time derivative blows up in finite time - a behavior well-documented in continuous parabolic models but increasingly complex in discrete settings. We employ a finite-difference scheme to discretize the governing equations. A primary result of this work is the proof that discrete quenching always occurs whenever the continuous counterpart quenches. We also examine the convergence of the discrete quenching time to its continuous counterpart as the mesh size approaches zero. These results offer critical insights into the reliability of finite-difference methods for capturing singular behaviors in non-local nonlinear problems, ensuring that numerical artifacts do not obscure the underlying physical properties of the system.

Lifespan estimates of solutions for the semilinear parabolic equations

Berikbol Torebek Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):

Abstract: We will discuss lifespan estimates for blow-up solutions of semilinear parabolic equations on unbounded domains such as whole Euclidean space, half space and exterior domains. By using a suitable family of test functions, we obtain upper bounds for the lifespan of solutions. In some cases, we also derive lower bounds for the lifespan.

Lifespan of solutions to systems of semilinear wave equations in one space dimension

Kyouhei Wakasa Muroran Institute of Technology Japan Co-Author(s):    Soichiro Katayama

Abstract: In this talk, we consider the systems of semilinear wave equations with multiple speeds in one space dimensions. We obtain the almost optimal lifespan estimate under some condition weaker than the null condition. This is a joint work with Professor Soichiro Katayama (The University of Osaka).

Recent progress on the semilinear damped wave equation with slowly decaying data

Yuta Wakasugi Hiroshima University Japan Co-Author(s):    Yuta Wakasugi

Abstract: Consider the Cauchy problem for the semilinear damped wave equation $u_{tt} - \Delta u + u_t = |u|^p$ for $t>0, x\in \mathbb{R}^n$. By the studies conducted from the 1990s to the early 2000s, it is known that, for smooth compactly supported initial data, the critical exponent is given by the so-called Fujita critical exponent ($p=1+2/n$). Subsequently, the critical exponent problem with non-compactly supported or slowly decaying (in general not in $L^1$) initial data has also been studied. In this talk, we review previous studies and present some recent progress on this topic.

Physical space approach to bilinear estimates and applications to wave and dispersive equations

Yi Zhou Fudan University Peoples Rep of China Co-Author(s):

Abstract: We develop a new bilinear estimate method based on a new div curl lemma. We can use our method to give alternative proof of low regularity local well posedness for dispersive equations, which previously rely on Bourgain space. We can also use our method to give new proof of global existence of classical solutions for nonlinear wave equations with small initial data. Moreover, we establish new results including the proof of Weiyue Ding`s conjecture for periodic Schrodinger flow, the global well posedness in the critical Besov space of the skew mean curvature flow and the gloabl well posednee in critical Sobolev space of Ishimori equations.