In this paper, our main concern is the well-posedness of the initial boundary value problem for a linear heat equation with a time-dependent, strongly singular potential $V \in C([0,T]; L^{\frac{N}{2}} (\Omega))$: $$ \begin{cases} u_t -\Delta u = V u & \text{in~} (0,T) \times \Omega, \\ u = 0 & \text{on~} (0,T) \times \partial \Omega, \\ u(0,x) =u_0(x) & \text{in~} \Omega, \end{cases} $$ where $u_0$ is initial data in $L^p(\Omega)$, $p \geq 1$. We show that the problem is well-posed on $L^p(\Omega)$, $p > 1$ within some appropriate class of solutions, and in turn the well-posedness breaks down on $L^1(\Omega)$. Furthermore, we also present some nonuniqueness results for the time-bounded potential class $L^\infty (0,T; L^{\frac{N}{2}} (\Omega))$.

We discuss the linearized equations of fiber spinning in the nonisothermal viscous regime. We prove that the solutions of the governing equations are generated by an eventually compact semigroup of bounded linear operators. This result gives us the justification to relate the stability of the semigroup to the numerical resolution of the spectrum of its generator. We report several numerical findings, confirming prior results on the stability of solutions for the equations of melt-spinning.

We investigate the existence of principal eigenvalues, i.e., values of $\lambda$ for which the corresponding eigenfunction is positive, for the problem $ - \Delta u(x) = \lambda g(x) u(x)$ for $x \in \mathbb R ^N$ where $g$ is a smooth function which may change sign. Unlike most previous studies the eigenfunction is not required to $\to 0$ as $|x| \to \infty$. It is shown that there may exist a closed interval of principal eigenvalues $[\lambda_* , \lambda^*]$ and sufficient conditions are given to ensure that principal eigenfunctions $\to 0$ as $|x| \to \infty$ if and only if $\lambda = \lambda^*$ or $\lambda_*$.

We prove a global existence and gradient estimates of solutions to the initial-boundary problem of the quasilinear parabolic equation \[ u_t - \mbox{div}\{\sigma(|\nabla u|)^2 \nabla u\} + g(\nabla u) =0 \mbox{ in } \Omega \times (0,\infty),\] with the initial and boundary conditions $ u(0,x)=u_0(x), u|_{\partial\Omega}=0$, where $ \Omega $ is a bounded domain in $ R^N , \sigma(v)$ is a function like $ \sigma(v)=1/\!\sqrt{1+v}$ and $g(\nabla u)$ is a nonlinear perturbation like $ g(\nabla u)\!=\pm|\nabla u|^{\alpha +1}\!,$ $ \alpha >0$. In particular, we derive the estimate \[ ||\nabla u(t)||_{\infty} \leq C(||\nabla u_0||_{p_0})t^{-N/(2p_0-3N)}e^{-\lambda t}, t >0\] for a certain $\lambda > 0 $, under the assumptions that $||\nabla u_0||_{p_0}, p_0 >3(N+\alpha) \quad ( p_0 > 2\alpha+5 \mbox{ if } N=1) $, is small and the mean curvature of the boundary $ \partial \Omega $ is nonpositive.

For $p>1$ we prove a compactness result for $p$-harmonic maps with values in $S^k$, the $(k+1)$-dimensional sphere. We generalize a lemma from [12] to vector-valued functions with assumptions on the $p$-Laplacian. We obtain the existence of weak solutions of the $p$-harmonic flow with values in $S^k$ for each $k\geq 1$ and $p>1$.

The linear viscoelastic equation is considered. We prove existence and uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary and provided the relaxation function decays exponentially. The existence is proved by means of the Faedo-Galerkin method, and the asymptotic behaviour is obtained by making use of the multiplier technique combined with integral inequalities due to Komornik.