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We consider the nonlinear Schrodinger equation on ${mathbb R}^N $, $Nge 1$, begin{equation*} partial _t u = i Delta u + lambda | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} end{equation*} with $lambda in {mathbb C}$ and $Re lambda >0$, for $H^1$-subcritical nonlinearities, i.e. $alpha >0$ and $(N-2) alpha < 4$. Given a compact set $K subset {mathbb R}^N $, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K $ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( Re lambda )^{- frac {1} {alpha }} (-alpha t + A(x) )^{ -frac {1} {alpha } - i frac {Im lambda } {alpha Re lambda } }$, where $Age 0$ vanishes exactly on $ K $, which is a solution of the ODE $u= lambda | u |^alpha u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4].
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We consider the nonlinear Schrodinger equation [ u_t = i Delta u + | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} ] for $H^1$-subcritical or critical nonlinearities: $(N-2) alpha le 4$. Under the additional technical assumptions $alphageq 2$ (and thus $Nleq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^alpha U$. In the simplest case, $U(t,x)=(|x|^k-alpha t)^{-frac 1alpha}$ for $t<0$, $xin {mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|Delta U|ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.
We consider the energy critical semilinear heat equation $$ left{begin{aligned} &partial_t u-Delta u =|u|^{frac{4}{n-2}}u &mbox{in } {mathbb R}^ntimes(0,T), &u(x,0)=u_0(x), end{aligned}right. $$ where $ ngeq 3$, $u_0in L^infty({mathbb R}^n)$, and $Tin {mathbb R}^+$ is the first blow up time. We prove that if $ n geq 7$ and $ u_0 geq 0$, then any blowup must be of Type I, i.e., [|u(cdot, t)|_{L^infty({mathbb R}^n)}leq C(T-t)^{-frac{1}{p-1}}.] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).
We consider the nonlinear heat equation with a nonlinear gradient term: $partial_t u =Delta u+mu| abla u|^q+|u|^{p-1}u,; mu>0,; q=2p/(p+1),; p>3,; tin (0,T),; xin R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|log(T-t)|^{1/2},$ like in the standard nonlinear heat equation, i.e. $mu=0,$ but as $(T-t)^{1/2}|log(T-t)|^{beta}$ with $beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $ abla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.
Let $Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=varphi(t)f(x,y)$ with $varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonovs regularization and truncated integration, we construct the regularized solutions. Numerical part is given.
We consider the focusing energy subcritical nonlinear wave equation $partial_{tt} u - Delta u= |u|^{p-1} u$ in ${mathbb R}^N$, $Nge 1$. Given any compact set $ E subset {mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = kappa (t + A(x) )^{ -frac {2} {p-1} }$, where $Age 0$ vanishes exactly on $ E$, which is a solution of the ODE $h = h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen $A$), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.
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