We prove existence in arbitrary space dimension of nonnegative bounded weak solutions of both the Cauchy problem and an initial-boundary value problem for a degenerate parabolic system which arises in plasma physics.

For a semilinear second order differential equation on $\mathbb{R}$, the existence of a continuous curve of positive solutions which bifurcates from the lowest eigenvalue of the linearized problem is proved. This curve can be parameterized globally by $\lambda$ and can be extended to infinity. We establish that all solutions of the equation are even and monotone, and under appropriate conditions, all of them belong to the curve of bifurcation. Our results depend heavily on the combination of symmetry and monotonicity imposed on the equation.

We present two different existence and uniqueness algorithms for constructing global mild solutions in $\mathcal {C}([0,T); L^3(\mathbb{R}^3))$ to the Cauchy problem for the Navier-Stokes equations with an external force.

We consider the quasilinear differential equation $ u_t = \mbox{ div} (|\bigtriangledown u|^{m-1} \bigtriangledown u )$ in $R^n \times (0, \infty)$, where $ 0 <m < 1$, $ n \geq 2$. We show that there exist various global self-similar solutions with different decaying rate when $ |x| \rightarrow \infty$. In particular, when $ (n-1)/(n+1) < m < 1$, for every $ c > 0$, there exists a unique self-similar solution which has the property that $u(x,t) \rightarrow c \delta (x)$ as $ t \rightarrow 0$, where $ \delta (x)$ is the Dirac delta function (Theorem 1). On the other hand, if $ 0 < m <(n-1)/(n+1)$, we show that there exist global self-similar solutions $u_{gs}$ which satisfy $u_{gs}(\cdot, t) \in L^p(R^n)$ for any $ p > s \equiv n(1-m)/(1+m)$ but they are not in $L^s(R^n)$ (Theorem 1). At the same time, we demonstrate the existence of finite time extinction self-similar solutions $u_{ls} $ which satisfy the same integrability conditions (Theorem 2).

In this paper we study the global existence of non-negative solutions to the Cauchy problem for $Lu= - u^{p}$ where $L$ belongs to a class ${\cal L}$ of hypoelliptic operators of degenerate parabolic type and $p>1$. Extending some old results by Fujita, we prove the existence and we determine explicitly a critical exponent $p^{*}$ for the problem. Namely, we prove that if $p>p^{*}$ then there are global positive solutions to the problem, while if $1 <p<p^{*}$ then all non-trivial solutions blow up in finite time. We also study the critical case $p=p^{*}$ for a remarkable subclass of ${\cal L}$.