We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧$\partial (e^{\beta u}-1)/\partial t-{div\left(\right|\nabla u|}^{p-2}\nabla u)+div(c(x,t){\left|u\right|}^{s-1}u)+b(x,t){\left|\nabla u\right|}^r=f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{\beta u}-1)(x,0)=(e^{\beta u₀}-1)\left(x\right)$ in Ω. with s = (N+2)/(N+p) (p-1), $c(x,t)\in {\left(L^{\tau }\left(QT\right)\right)}^N$, τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b(x,t)\in L^{N+2,1}\left(QT\right)$ and f ∈ L¹(Q).

Consider the nonlinear heat equation (E): $u_t-\Delta u={\left|u\right|}^{p-1}u+b{\left|\nabla u\right|}^q$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C₁{(T-t)}^{-1/(p-1)}{\le \left|\right|u\left(t\right)\left|\right|}_{\infty }\le C₂{(T-t)}^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_t-u_{xx}\ge u^p$. More general inequalities of the form $u_t-u_{xx}\ge f\left(u\right)$ with, for instance, $f\left(u\right)=(1+u)lo{g}^p(1+u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_t\ge -{(-\Delta )}^{\beta /2}u-V\left(x\right)u+h(x,t)u^p$, where ${(-\Delta )}^{\beta /2}$, 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V₊\left(x\right)\sim a\left|x\right|}^{-b}$, where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

We provide a duality theory and existence results for a operator equation $\nabla T(x)=\nabla N(x)$ where $T$ is not necessarily a monotone operator. We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.

Let $u\in L^2(I;H^1\left(\Omega \right))$ with ${\partial }_{t}u\in L^2(I;H^1{\left(\Omega \right)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds ${\partial }_t{u}^{+}\notin L^1(I;H^1{\left(\Omega \right)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit....

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions...

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\cdots \&lt;{\lambda }_k^{-}\le {\lambda }_k^{+}\&lt;{\lambda }_{k+1}^{-}\le \cdots$ of a Zakharov-Shabat operator is symmetric,. ${\lambda }_k^{\pm }=-{\lambda }_{-k}^{\mp }$ for all $k$, if and only if the sequence ${\left({\gamma }_k\right)}_{k\in \mathbb{Z}}$ of gap lengths, ${\gamma }_k:={\lambda }_k^{+}-{\lambda }_k^{-}$, is symmetric with respect to $k=0$.

We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $-\Delta u=f_{\lambda }(x,u)$, $u\in H_0^1\left(\Omega \right)$, where $\Omega$ is a bounded domain in ${ℝ}^N$, $N\ge 3$ and $\lambda >0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $\lambda a\left(x\right)u^q+b\left(x\right)u^p$, where $0\le q<1<p\le 2^{*}-1$. The coefficient $a\left(x\right)$ is assumed to be nonnegative but $b\left(x\right)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, where the $u$-field and the $\xi$-field are $ℝ$-valued, $\kappa \in [0,\infty )$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $\xi$-field plays the role of athat drives the equation. The initial condition $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$,...

$\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, of class $C^2$ and $T>0$. In the cylinder $Q=\mathrm{\Omega }\times \left(0,T\right)$ we consider non variational basic operator $a\left(H\left(u\right)\right)-\partial u/\partial t$ where $a\left(\xi \right)$ is a vector in ${\mathbb{R}}^N$, $N\ge 1$, which is continuous in $\xi$ and satisfies the condition (A). It is shown that $\mathrm{\forall }f\in L^2\left(Q\right)$ the Cauchy-Dirichlet problem $u\in W_0^{2,1}\left(Q\right)$, $a\left(H\left(u\right)\right)-\partial u/\partial t=f$ in $Q$, has a unique solution. It is further shown that if $u\in W_0^{2,1}\left(Q\right)$ is a solution of the basic system $a\left(H\left(u\right)\right)-\partial u/\partial t=0$ in $Q$, then $H\left(u\right)$ and $\partial u/\partial t$ belong to $H_{loc}^1\left(Q\right)$. From this the Hölder continuity in $Q$ of the vectors $u$ and $Du$ are deduced respectively when $n\le 4$ and...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $li{m}_{n\to \infty }{n}^{-1}log{\sum }_{j=0}^{\tau ₙ\left(x\right)}\mu \left(\alpha ⁿ\left(T^j\left(x\right)\right)\right)$, where $\alpha ⁿ\left(T^j\left(x\right)\right)$ is the element of the partition containing $T^j\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega$ whose boundary $\partial \Omega$ is formed by two circles ${\Gamma }_1$, ${\Gamma }_2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho$, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^{-1,\alpha }(\mathrm{\Omega })$ consinsting of all derivatives of hölder-continuous functions in $\mathrm{\Omega }$ where $\mathrm{\Omega }$ is a domain in ${ℝ}^n$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^{-1,\alpha }(\mathrm{\Omega })$. We prove also that the spaces $C^{-1,\alpha }(\mathrm{\Omega })$ can be considered as extrapolation spaces relative to suitable non-variational...