Related papers: Gevrey well posedness for $3$-evolution equations …
This paper mainly investigates the Cauchy problem of the spatially weighted dissipative equation with initial data in the weighted Lebesgue space. A generalized Hankel Transform is introduced to derive the analytical solution and a special…
We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of…
We consider the global well-posedness for the Cauchy probelem of the Kawahara equation which is one of the fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global…
We investigate global well-posedness to the Cauchy problem of three-dimensional compressible viscous and heat-conducting micropolar fluid equations with zero density at infinity. By delicate energy estimates, we establish global existence…
We consider solutions of the Cauchy problem for semilinear equations with (possibly) different L\'evy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the…
We study the well-posedness of the initial value problem on periodic intervals for linear and quasilinear evolution equations for which the leading-order terms have three spatial derivatives. In such equations, there is a competition…
In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension $d\geq2$. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contain a Navier-Stokes…
We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the…
In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through…
In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws…
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right.…
In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Gr\"{u}nrock's result in 3D, we take the data in the Fourier-Lebesgue spaces…
In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated to a family equations of ZK-KP-type \[ \begin{cases} u_{t}=u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0)=\psi \in Z \end{cases} \]…
This paper considers the two-dimensional Cauchy problem of the full compressible Navier-Stokes equations with far-field vacuum in $\mathbb{R}^2$, where the viscosity and heat-conductivity coefficients depend on the absolute temperature…
We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such…
We investigate sufficient H\"older continuity conditions on Leray-Hopf (weak) solutions to the in unsteady Navier-Stokes equations in three dimensions guaranteeing energy conservation. Our focus is on the half-space case with homogeneous…
We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the…
We consider evolution equations for two classes of generalized anharmonic oscillators and the associated initial value problem in the space of tempered distributions. We prove that the Cauchy problem is well posed in anisotropic…
In this paper we investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}%…
We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of…