Papers

This paper investigates an initial-Neumann boundary value problem for a Keller–Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $\gamma(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.

This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined problems for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the $C$-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{Barkatou2002}. We provide a new, self-contained geometric result: for almost every $t$, the level sets of the solution inherit the $C$-GNP property. This inheritance theorem justifies the variational framework and guarantees that the entire foliation generated by the state function remains within the admissible class. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. The physical interpretation of the bi-Laplacian problem $\mathcal{B}(f,g)$ in the Kirchhoff-Love theory of thin plates is emphasized.

We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term.