arXiv subjects

Aleksander Cwiszewski

Publications and source records attributed to Aleksander Cwiszewski.

6 recordsLinked to original sources

arXiv2017

Forced periodic solutions for nonresonant parabolic equations on R^N

Criteria for the existence of $T$-periodic solutions of nonautonomous parabolic equation $u_t = Δu + f(t,x,u)$, $x\in\mathbb{R}^N$, $t>0$ with asymptotically linear $f$ will be provided. It is expressed in terms of time average function $\hat f$ of the nonlinear term $f$ and the spectrum of the Laplace operator $Δ$ on $\mathbb{R}^N$. One of them says that if the derivative $\hat f_\infty$ of $\hat f$ at infinity does not interact with the spectrum of $Δ$, i.e. $\mathrm{Ker} (-Δ+ \hat f_\infty)=\{0\}$, then the parabolic equation admits a $T$-periodic solution. Another theorem is derived in the situation, where the linearization at $0$ and infinity differ topologically, i.e. the total multiplicities of negative eigenvalues of the averaged linearizations at $0$ and $\infty$ are different mod $2$.

math.AP↗

arXiv2017

Continuity and averaging for parabolic evolution systems

Averaging principle for abstract non-autonomous parabolic evolution equations governed by time-dependent family of positive sectorial operators is proved. Apart from linear case also a nonlinear version for continuous perturbations is provided. These results extend Henry's averaging technique to the case where linear part of the equation depends on time and results due to Ilyin that were for dissipative equations.47D06, 34G05, 34G20

math.FA↗

arXiv2016

Stationary solutions and connecting orbits for $p$-Laplace equation

We deal with one dimensional $p$-Laplace equation of the form $$ u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, $$ under Dirichlet boundary condition, where $p>2$ and $f\colon [0,l]\times \mathbb{R}\to \mathbb{R}$ is a continuous function with $f(x,0)=0$. We will prove that if there is at least one eigenvalue of the $p$-Laplace operator between $\lim_{u\to 0} f(x,u)/|u|^{p-2}u$ and $\lim_{|u|\to +\infty} f(x,u)/|u|^{p-2}u$, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are obtained by use of Conley type homotopy index and continuation along $p$ techniques. We obtain stronger results than by use of fixed point index and additionally get the existence of a connecting orbit.

math.AP↗

arXiv2014

A Landesman-Lazer type result for periodic parabolic problems on $\mathbb{R}^N$ at resonance

We are concerned with $T$-periodic solutions of nonautonomous parabolic problem of the form $u_t = Δu + V(x) u + f(t,x,u)$, $t >0$, $x \in \mathbb{R}^N$, with $V \in L^\infty (\mathbb{R}^N)+L^p(\mathbb{R}^N)$, $p \geq N$ and $T$-periodic continuous perturbation $f:\mathbb{R}^N\times \mathbb{R} \to \mathbb{R}$. The so-called resonant case is considered, i.e. when ${\cal N}:=\mathrm{Ker} (Δ+ V) \neq \{0\}$ and $f$ is bounded by a square-integrable function. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time average mapping $\hat f: {\cal N}\to {\cal N}$ being the restriction of $f$ to ${\cal N}$. By use of the formula and continuation techniques we show that Landesman-Lazer type conditions imply the existence of $T$-periodic solutions.

math.AP↗

arXiv2012

Positive stationary solutions for p-Laplacian problems with nonpositive perturbation

The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian. We provide a general topological degree that detects solutions of the problem $$ \{{array}{l} A(u)=F(u) u\in M {array}. $$ where $A:X\supset D(A)\to X^*$ is a maximal monotone operator in a Banach space $X$ and $F:M\to X^*$ is a continuous mapping defined on a closed convex cone $M\subset X$. Next, we apply this general framework to a class of partial differential equations with $p$-Laplacian under Dirichlet boundary conditions.

math.AP↗

arXiv2011

Averaging principle and hyperbolic evolution equations

An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry's results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.

math.AP↗