Related papers: On variable Lebesgue spaces and generalized nonlin…
This paper is concerned with the Cauchy problem of a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of ``heat'' or ``particles'' on the vertices of hypergraph. We consider the…
This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which…
This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup…
This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic…
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability…
In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~…
We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general.…
This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild…
We establish the global existence and uniqueness of $L^1$-solutions to the Cauchy problem for time-fractional porous medium type nonlinear diffusion equations. Furthermore, we give the mass conservation law for $L^1$-solutions to…
We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the…
In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\_t -\Delta u= u^{3 },\ u(0,x)=u\_0$. After extending Y. Meyer's result establishing the existence of global solutions, under a smallness condition of the…
We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about…
Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori…
In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under mild conditions on the coefficient of the principal part and without upper growth…
This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided…
In this paper, we consider the Cauchy problem for a class of weakly dissipative Camassa-Holm equations in nonhomogeneous Besov spaces. First, we prove that the Cauchy problem admits a unique global strong solution in Besov spaces with…
We study the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and…
We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…
We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a…