Christoph Walker
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Analytic semigroups in weighted L1-spaces on the half-line generated by singular or degenerate operators Open
Ranges of the real-valued parameters α, a, b, and m are identified for which the operator Aα(a,b)f(x):=xα(f′′(x)+axf′(x)+bx2f(x)),x>0, generates an analytic semigroup in L1((0,∞),xmdx).
On the principle of linearized stability for quasilinear evolution equations in time‐weighted spaces Open
Quasilinear (and semilinear) parabolic problems of the form with strict inclusion of the domains of the function and the quasilinear part are considered in the framework of time‐weighted function spaces. This allows one to establish the pr…
A potential theory approach to the capillarity-driven Hele-Shaw problem Open
In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This …
Analysis of a one-dimensional biofilm model Open
In this paper a reduced one-dimensional moving boundary model is studied that describes the evolution of a biofilm driven by the presence of a reaction limiting substrate. Global well-posedness is established for the resulting parabolic fr…
On the principle of linearized stability for quasilinear evolution equations in time-weighted spaces Open
Quasilinear (and semilinear) parabolic problems of the form $v'=A(v)v+f(v)$ with strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the function $v\mapsto f(v)$ and the quasilinear part $v\mapsto A(v)$ are consi…
Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities Open
It is shown that semilinear parabolic evolution equations featuring Hölder continuous nonlinearities with at most linear growth possess global strong solutions for a general class of initial data. The abstract results are applied to a rece…
Well-posedness of quasilinear parabolic equations in time-weighted spaces Open
Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the…
Recovering Initial States in Semilinear Parabolic Problems from Time-Averages Open
Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of superlin…
On a quasilinear parabolic–hyperbolic system arising in MEMS modeling Open
A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the …
Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities Open
It is shown that semilinear parabolic evolution equations $u'=A+f(t,u)$ featuring Hölder continuous nonlinearities $ f=f(t,u)$ with at most linear growth possess global strong solutions for a general class of initial data. The abstract res…
A note on the compactness of the resolvent of the age-diffusion operator Open
The generator of the semigroup associated with linear age-structured population models including spatial diffusion is shown to have compact resolvent.
Stability and Instability of Equilibria in Age-Structured Diffusive Populations Open
The principle of linearized stability and instability is established for a classical model describing the spatial movement of an age-structured population with nonlinear vital rates. It is shown that the real parts of the eigenvalues of th…
On a Quasilinear Parabolic-Hyperbolic System Arising in MEMS Modeling Open
A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the …
Well-Posedness of Quasilinear Parabolic Equations in Time-Weighted Spaces Open
Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the…
Well-posedness and stability analysis of an epidemic model with infection age and spatial diffusion Open
A compartment epidemic model for infectious disease spreading is investigated, where movement of individuals is governed by spatial diffusion. The model includes infection age of the infected individuals and assumes a logistic growth of th…
Well-posedness of the coagulation-fragmentation equation with size diffusion Open
Local and global well-posedness of the coagulation-fragmentation equation\nwith size diffusion are investigated. Owing to the semilinear structure of the\nequation, a semigroup approach is used, building upon generation results\npreviously…
The fragmentation equation with size diffusion: Well posedness and long-term behaviour Open
The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator.…
Energy minimizers for an asymptotic MEMS model with heterogeneous dielectric properties Open
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. Thi…
Properties of the Semigroup in $L_1$ Associated with Age-Structured Diffusive Populations Open
The linear semigroup associated with age-structured diffusive populations is investigated in the $L_1$-setting. A complete determination of its generator is given along with detailed spectral information that imply, in particular, an async…
Convergence of Energy Minimizers of a MEMS Model in the Reinforced Limit Open
Energy minimizers to a MEMS model with an insulating layer are shown to converge in its reinforced limit to the minimizer of the limiting model as the thickness of the layer tends to zero. The proof relies on the identification of the $\Ga…
The fragmentation equation with size diffusion: Well-posedness and long-term behavior Open
The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in (0, $\infty$). The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schr{ö}dinger operato…
The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions Open
The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponenti…
Convergence of energy minimizers of a MEMS model in the reinforced limit Open
Energy minimizers to a MEMS model with an insulating layer are shown to\nconverge in its reinforced limit to the minimizer of the limiting model as the\nthickness of the layer tends to zero. The proof relies on the identification of\nthe $…
Strong solutions to a nonlocal-in-time semilinear heat equation Open
The existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. …
Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model Open
Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses …
Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties Open
The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system wit…