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...

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ${\partial }_{t}u+{div}_{y}A(y,u)-{\Delta }_{y}u=0$, where $y\in {ℝ}^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^1$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

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.

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 consider a nonlinear parabolic system modelling chemotaxis $u_t=\nabla ·(\nabla u-u\nabla v)$, $v_t=\Delta v+u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

We study the existence of solutions of the nonlinear parabolic problem $\partial u/\partial t-div\left[\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right))]+\alpha \left(u\right)=f$ in ]0,T[ × Ω, $\left(\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right)))·\eta +\gamma \left(u\right)=g$ on ]0,T[ × ∂Ω, u(0,·) = u₀ in Ω, with initial data in L¹. We use a time discretization of the continuous problem by the Euler forward scheme.

Large time behavior of solutions to the generalized damped wave equation $u_{tt}+A{u}_t+\nu Bu+F(x,t,u,u_t,\nabla u)=0$ for $(x,t)\in {ℝ}^n\times [0,\infty )$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A{u}_t=u_t$, $Bu=-\Delta u$, and the nonlinear term is either $|u_t{|}^{q-1}{u}_t$ or ${\left|u\right|}^{\alpha -1}u$. In this case, the asymptotic profile of solutions...

An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{\alpha }$. This semigroup possesses an $(X^{\alpha }-Z)$-global attractor that is closed, bounded, invariant in $X^{\alpha }$, and attracts bounded subsets of $X^{\alpha }$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...

We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=-{\left(-\Delta \right)}^{m}u\pm {\left|u\right|}^{p-1}u$ in ${ℛ}^N\times {ℝ}_{+}$ $(p>1)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<p_0=1+2m/N$, the Cauchy problem admits a solution $u(x,t)$ which is bounded and smooth for small $t>0$, while for $p\ge p_0$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

We show in two dimensions that if $Kf={\int }_{ℝ₊²}k(x,y)f\left(y\right)dy$, $k(x,y)=\left(e^{i{x}^a·y^b}\right){/\left(\right|x-y|}^{\eta })$, p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_p\left(y\right)=y^{(p/p^{\text{'}})(1̅-b/a)}$, then ${\left|\right|Kf\left|\right|}_p{\le C\left|\right|f\left|\right|}_{p,v_p}$ if η + α₁ + α₂ < 2, ${\alpha }_j=1-b_j/a_j$, j = 1,2. Our methods apply in all dimensions and also for more general kernels.

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1<p<2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t\to \infty$.

We prove the existence of global attractors for the following semilinear degenerate parabolic equation on ${ℝ}^N$: ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y)=px+(1-p)y-{\sum }_{s=m}^{\infty }s(x,y)$ uniformly in a neighborhood of the origin, where m > 1, ${}_s(x,y)={\sum }_{i=0}^s{a}_{i,s}{x}^{s-i}{y}^i$; (c) $ₘ(1,1)={\sum }_{i=0}^m{a}_{i,m}>0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1}=f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

We consider the initial-value problem for a linear hyperbolic parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove $L^p-L^q$ time decay estimates for the solution of the associated linear Cauchy problem.

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f\left(u\right)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that the omega limit set $\omega \left(u\right)$ of every bounded positive solution is determined by a stationary...

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first...

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $\epsilon cₜ=k_c\Delta c-f\left(c\right)c+g(a,c)$, x ∈ Ω, t > 0, for $\Omega \subset {ℝ}^N$, completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.