Nonlinear parabolic equations, relaxation and roughness

In a recent paper, Aubin and Coulouvrat (1998) dealt with equations of motion in the fluid in relaxation - mathematically: Burgers' equations, physically: continuity equations, Navier-Stokes equations, energy bilance, and equations of relaxation. A related equation of Ginzburg and Landau type (1965) extended for the kinetic depinning transitions takes its form

(0)     

Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions

This paper is concerned with the following one-dimensional nonlinear system of equations:

(0.1)     

Zero relaxation limit to centered rarefaction waves for Jin-Xin relaxation system

In this paper, we study the zero relaxation limit problem for the following Jin-Xin relaxation system

(E)     

Global decaying solution to dissipative nonlinear evolution equations with ellipticity

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations:

(E)     

Exponential decay for Maxwell equations with a boundary memory condition

We study the asymptotic behavior of the solution of the Maxwell equations with the following boundary condition of memory type:

(0.1)     

Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states

In this paper, we consider the global existence and asymptotic behaviors of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

(E)     

Smooth solution for the porous medium equation in a bounded domain

In this paper, we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain:

(0.1)     

Infinite horizon BSDEs with dissipative coefficients in Hilbert spaces and applications

In this paper we study a class of infinite horizon backward stochastic differential equations (BSDEs) of the form

     

Convergence to diffusion waves for nonlinear evolution equations with different end states

In this paper, we consider the global existence and the asymptotic decay of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

(E)     

KAM tori of Hamiltonian perturbations of 1D linear beam equations

In this paper, one-dimensional (1D) nonlinear beam equations

     

On the optimal control of systems described by implicit evolution equations

We obtain conditions for the existence and uniqueness of an optimal control for the linear nonstationary operator-differential equation

$\frac{d}{{dt}}[A(t)y(t)] + B(t)y(t) = K(t)u(t) + f(t)$

with a quadratic performance criterion. The operators A(t) and B(t) are closed and may have nontrivial kernels. The results are applied to differential-algebraic equations and to partial differential equations that do not belong to the Cauchy-Kowalewskaya type.

     

L-convergence rate to diffusion waves for p-system with relaxation

In this paper, we consider the asymptotic behavior of solution for the Cauchy problem for p-system with relaxation

(E)     

Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

We are interested in the following class of equations:

     

Some remarks on the classification of global solutions with asymptotically flat level sets

We simplify some technical steps from Savin (Ann Math. (2) 169(1):41–78, 2009) in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type

$$\begin{aligned} \triangle u=W^{\prime }(u), \end{aligned}$$

where W is a double well potential.

     

Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

We study the global solvability of the Cauchy-Dirichlet problem for two second order in time nonlinear integro-differential equations:

1)

the extensible beam/plate equation

     

Subcritical nonlinear nonlocal equations on a half-line

We study nonlinear nonlocal equations on a half-line in the subcritical case

(0.1)     

On the alienation of the logarithmic and exponential Cauchy equations

In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107–121, 2016) in connection with the functional equation

$$\begin{aligned} f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y). \end{aligned}$$

Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien.

     

Positive periodic solutions of higher-dimensional nonlinear functional difference equations

By using a well-known fixed point index theorem, we study the existence, multiplicity and nonexistence of positive T-periodic solution(s) to the higher-dimensional nonlinear functional difference equations of the form

     

Existence results for nonlinear fractional differential equations in <Emphasis Type="Italic">C</Emphasis>[0, <Emphasis Type="Italic">T</Emphasis>)

In this paper, we investigate the existence results for fractional differential equations of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$

(0.1)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$

(0.2)

where \(D_{c}^{q}\) is the Caputo fractional derivative. We prove the above equations have solutions in C[0, T). Particularly, we present the existence and uniqueness results for the above equations on \([0,+\infty )\).