arXiv subjects

Renata Lukasiak

Publications and source records attributed to Renata Lukasiak.

3 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↗

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↗