相关论文: Continuous and discrete one dimensional autonomous…

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-\tau)u(\tau)\,d\tau -k(t)u(0)$ where $k$ is a…

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix.…

We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal…

In this paper we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation $-\Delta u=e^u~~~{in }~~~\Omega$, where $\Omega$ is an open set of $R^3$. By improving the partial regularity…

In this paper, we investigate the well-posedness and the long-time asymptotic behavior for the initial-boundary value problem for multi-term time-fractional diffusion equations, where the time differentiation consists of a finite summation…

Let $f_1,f_2$ be linearly independent solutions of $f''+Af=0$, where the coefficient $A$ is an analytic function in the open unit disc $\mathbb{D}$ of $\mathbb{C}$. It is shown that many properties of this differential equation can be…

The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and…

For $0<\alpha<\frac{1}{3}$ we construct unique solutions to the fractal Burgers equation $\partial_t u + u\partial_xu + (-\Delta)^\alpha u = 0$ which develop a first shock in finite time, starting from smooth generic initial data. This…

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,\tau)||y_\tau||+\int_\tau^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+\rho, \hspace{0.5cm}…

This work considers to numerically solve a subdiffusion equation involving constant time delay $\tau$ and Riemann-Liouville fractional derivative. First, a fully discrete finite element scheme is developed for the considered problem under…

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))\ \mbox{in}\ \Omega \newline u\geq 0\ \mbox{in}\…

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique,…

We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several…

This contribution investigates the computational complexity of simulating linear ordinary differential equations (ODEs) on digital computers. We provide an exact characterization of the complexity blowup for a class of ODEs of arbitrary…

We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is…

The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in…

We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected.…

The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in…

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For…