In this paper, we study almost sure exponential stability and asymptotic stability in probability of mild solutions of stochastic partial functional differential equations in Hilbert spaces. The linear part is assumed to generate an analytic semigroup while the drift and diffusion terms satisfy a Hölder type condition with respect to the fractional power norm of the linear part.

We consider weak positive solutions of the equation $-\Delta_m u=f(u)$ in the half-plane with zero Dirichlet boundary conditions. Assuming that the nonlinearity $f$ is locally Lipschitz continuous and $f(s)>0$ for $s>0$, we prove that any solution is monotone. Some Liouville-type theorems follow in the case of Lane-Emden-Fowler-type equations. Assuming also that $|\nabla u|$ is globally bounded, our result implies that solutions are one dimensional, and the level sets are flat.

We obtain multiple positive solutions of singular semilinear elliptic boundary-value problems using a perturbation method and critical point theory.

Reaction-convection-diffusion equations with a monostable reaction term, that generalize the KPP equation, admit a global travelling-wave solution whose limiting values are the stable and unstable steady states if and only if the wave-speed is greater than or equal to some critical number. In a recent paper, Crooks and Mascia showed that, in the vanishing viscosity limit, this minimal speed tends to the corresponding minimal wave-speed associated with the first-order equation without the diffusion term. An alternative proof of this result is presented using an integral equation approach developed by the author and Robert Kersner.

In this paper, we consider a drift-diffusion model of parabolic-elliptic type, with three coupled equations. We prove that there exist parameter regimes for which non-simultaneous blow-up of solutions happens. This is in contrast to a two-chemotactic species model, coupled to an elliptic equation for an attractive chemical produced by the two species, where blow-up of one species implies blow-up of the other one at the same time. Also, we show that the range of parameters of the drift-diffusion model in this paper, for which blow-up happens, is larger than suggested by previous results in the literature.

The Cauchy problem for 1-D nonlinear Schrödinger equations with quadratic nonlinearities are considered in the spaces $H^{s,a}$ defined by $ \| f \|_{H^{s,a}}=\| (1+|\xi|)^{s-a} |\xi|^a \widehat{f} \|_{L^2}, $ and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term $u\bar{u}$. In particular, when $a=0$ the previous well-posedness result in $H^s$, $s>-1/4$, given by Kenig, Ponce and Vega (1996), is improved to $s\ge -1/4$. This also extends the result in $H^{s,a}$ by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the ``good'' Boussinesq equation and yields local well-posedness in $H^s\times H^{s-2}$ with $s>-1/2$, which is an improvement of the previous result given by Farah (2009).

We prove the vanishing-viscosity $L^1({\mathbb{R}})$-convergence of minimal-speed travelling-front profiles for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$. Such equations are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both for $\varepsilon >0$, when all fronts are smooth, and for $\varepsilon =0$, when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold. Our main result is that, as $\varepsilon \downarrow 0$, profiles of minimal velocity ${c_{\varepsilon}^*}$ converge in $L^1({\mathbb{R}})$ to the unique (up-to-translation) $\varepsilon=0$ entropy-front profile of minimal velocity ${c^*}$. The proofs exploit the compactness inherent in the monotonicity of the profiles, together with uniform-in-$\varepsilon$ estimates on the convergence of the profiles to their spatial limits. Convergence results for the less-delicate fronts of non-minimal speed also follow from the arguments given.

We consider the generalized Korteweg-de Vries equation $$ \partial_t u +\partial_x^3 u +\partial_x(u^p)=0, \quad (t,x)\in{\mathbb{R}}^2, $$ in the supercritical case $p>5$, and we are interested in solutions which converge to a soliton in large time in $H^1$. In the subcritical case ($p < 5$), such solutions are forced to be exactly solitons by variational characterization [3, 32], but no such result exists in the supercritical case. In this paper, we first construct a ``special solution" in this case by a compactness argument, i.e., a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton [20], we construct a one-parameter family of special solutions which characterizes all such special solutions. In the case of the nonlinear Schrödinger equation, a similar result was proved in [7, 8].

We study the Cauchy problem of the Schrödinger-Korteweg-de Vries system. First, we establish local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show that they are sharp in some well-posedness thresholds. In particular, we obtain local well posedness for the initial data in $H^{-\frac{3}{16}+}({\mathbb{R}})\times H^{-\frac{3}{4}+}({\mathbb{R}})$ in the resonant case; it is almost optimal except for the endpoint. Finally, we establish global well-posedness results in $H^s({\mathbb{R}})\times H^s({\mathbb{R}})$ when $s>\frac{1}{2}$ regardless of whether we are in the resonant case or in the non-resonant case, which improves the results of Pecher (2005).