Article
Mathematics
Zehan Dong, Nangao Zhang, Changjiang Zhu
Summary: In this paper, the Cauchy problem for a quasi-linear hyperbolic-parabolic chemotaxis system modelling vasculogenesis is investigated. Through the use of new correction functions and existing results, more general results are obtained.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Andre H. Erhardt
Summary: This study investigates the stability of a unique weak solution to certain parabolic systems with nonstandard growth conditions and cross-diffusion terms, demonstrating that two unique weak solutions with different initial values are controlled by these initial values.
Article
Mathematics
Qingqing Liu, Hongyun Peng, Zhi-An Wang
Summary: This paper investigates a quasi-linear hyperbolic-parabolic system modeling vasculogenesis, showing the existence of a nonlinear diffusion wave under suitable structural assumptions on the pressure function. The study demonstrates that the solution of the system will locally and asymptotically converge to this wave if the wave strength is small. Additionally, using time-weighted energy estimates, it is further proven that the convergence rate of the nonlinear diffusion wave is algebraic.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Cun-Ming Liu, Hao-Nan Dou
Summary: This paper discusses the stability of a certain type of boundary value problem for quasilinear hyperbolic systems, proving global existence of solutions under certain conditions and obtaining error estimates between solutions with different boundary data, which has theoretical and practical significance.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Vladimir Vasilyev, Natalya Zaitseva
Summary: This paper studies the construction of explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables. By using the formal application of an integral transformation, classical solutions are obtained under certain conditions.
Article
Mathematics, Applied
Yafeng Li, Chunlai Mu, Qiao Xin
Summary: In this paper, we discuss a hyperbolic-parabolic system on a network. The global existence of solution to this problem with suitable the transmission conditions at interior is obtained by energy estimates. Moreover, for the case of acyclic network, we prove the existence and uniqueness of stationary solution to the system and show that the stationary solution provides asymptotic profiles for a class of global solutions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Emanuele Malagoli, Diego Pallara, Sergio Polidoro
Summary: We establish surface and volume mean value formulas for uniformly parabolic equations with coefficients of low regularity. Using these formulas, we prove the parabolic strong maximum principle and the parabolic Harnack inequality. Notably, our results are based solely on classical theory and our arguments are similar to those used in the original theory of harmonic functions. We provide two proofs, relying on two different formulations of the divergence theorem, one for sets with almost C-1 boundary and the other for sets with finite perimeter.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics
Luis Silvestre
Summary: This article presents some properties of solutions to parabolic equations and fully nonlinear uniformly parabolic equations. By constructing examples and using numerical computations, it proves certain properties of the solutions under specific conditions. These studies are of great importance for a deeper understanding and solving problems related to parabolic equations and nonlinear uniformly parabolic equations.
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE
(2022)
Article
Mathematics
Harsh Prasad, Vivek Tewary
Summary: We prove the local boundedness of variational solutions to the double phase equation under certain restrictions on s, s', p, q, and the non-negative function a(x, y).
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Hongjie Dong, Seick Kim, Sungjin Lee
Summary: We construct the fundamental solution of second order parabolic equations in non-divergence form by assuming that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case where the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Hamid El Bahja
Summary: This work focuses on the study of the sub-critical case of an anisotropic fully parabolic Keller-Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without any restriction on the size of the initial data.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Article
Mathematics
Gonzalo Galiano, Julian Velasco
Summary: This study proves that under certain conditions, the nonlocal diffusion problem can be transformed into a problem with local diffusion by rescaling the kernel function. It also provides a new proof of existence for solutions to the local diffusion problem.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Article
Mathematics, Applied
Giuseppe Coclite, Lorenzo di Ruvo
Summary: The paper discusses the fifth-order short pulse equation modeling the nonlinear propagation of optical pulses, particularly circularly and elliptically polarized few-cycle solitons in a Kerr medium, and proves the well-posedness of classical solutions for the associated Cauchy problem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Engineering, Multidisciplinary
Mubashir Qayyum, Amna Khan, Syed Tauseef Saeed, Ali Akgul, Muhammad Bilal Riaz
Summary: Parabolic equations are widely used in various fields such as chemical engineering, vibration theory, particle diffusion, and heat conduction. This article proposes a residual power series algorithm for higher order parabolic equations with variable coefficients in multiple dimensions, which provides closed-form solutions without linearization or perturbation. The algorithm has been tested on homogeneous and non-homogeneous parabolic models, demonstrating its validity and effectiveness for complex scenarios in engineering and sciences.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics
Jie Jiang, Philippe Laurencot
Summary: This article demonstrates the global existence and boundedness of classical solutions for a parabolic-elliptic chemotaxis system with local sensing, assuming that the motility function is unbounded at infinity. The cornerstone of the proof lies in deriving L∞-estimates for the second component of the system by employing various comparison arguments.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2023)