Local Well-Posedness for Semilinear NLS/NLW via Strichartz Contraction

Intuition Beginner

You have an equation that wants to predict the future of a wave, but the equation is tangled: the wave's rate of change depends on the wave itself, raised to a power. There is no formula that just hands you the answer. So you do the next best thing. You guess an answer, plug the guess into the right-hand side of the equation, turn the crank, and read off a new wave. Then you feed that new wave back in and turn the crank again. If each pass changes the wave less than the pass before, the waves close in on a single limit, and that limit is the true solution.

The whole game is making sure each pass shrinks the gap. The crank has two parts. The linear part is the free spreading you already understand, and its effect on accumulated size is controlled by the space-time bound from the previous units. The nonlinear part is where the wave gets multiplied by itself; that is the dangerous part, because multiplying can make things bigger. The trick is to run the crank only over a short stretch of time. Over a short window the nonlinear part has little room to grow, so the shrinking wins, the passes settle down, and you have proved a solution exists for at least that short window.

There is a number attached to each equation, fixed by the dimension of space and the power in the nonlinearity, that decides how the short-time window behaves. When the linear spreading is strong enough relative to the nonlinear growth, you can always find a window — that is the easy, forgiving case. When the two are exactly in balance, you can still win, but only if the starting wave is small; the window no longer helps on its own. When the nonlinear growth dominates, this method gives up entirely. That single number sorts every such equation into one of those three boxes.

The one-sentence takeaway: you build the solution by repeated guessing, each guess closer than the last, and you guarantee the guesses settle by running over a short enough time that the free spreading's space-time bound beats the nonlinear growth.

Visual Beginner

Picture a funnel of nested guesses. At the wide mouth you drop in your first crude guess for the wave. Just below it sits the second guess, produced by one turn of the crank — closer to the centre line. Below that the third, closer still, and so on down the funnel, each guess hugging the centre more tightly than the one above. The centre line the funnel narrows toward is the true solution. The funnel narrows because each turn of the crank moves you a fixed fraction of the remaining distance toward the centre; a fixed fraction of a shrinking gap shrinks geometrically, so the guesses pile up onto the centre line and stop moving.

The companion picture is the sorting dial. Imagine a single slider whose position is set by two things you turn in: how many dimensions space has, and how strong the self-multiplication is. Slide it one way and you land in the forgiving box where a short window always works. Slide it to the exact middle mark and you land in the balanced box where only small starting waves are allowed. Slide it the other way and you fall into the box where this whole repeated-guessing strategy stops working. The slider never moves on its own; you read off which box you are in the moment you fix the dimension and the power.

Worked example Beginner

We compute the deciding number for one concrete equation and read off which box it lands in, by hand.

Step 1. Pick the cubic equation in three space dimensions. "Cubic" means the wave is multiplied by itself three times in the nonlinear part, so the power $p$ is $3$. The dimension $n$ is $3$.

Step 2. Write down the deciding-number rule. The rule says: take the dimension, divide by two, and from it subtract two divided by the quantity (the power minus one). In symbols the deciding number is $n/2$ minus $2/(p-1)$.

Step 3. Plug in the numbers. The first piece is $n/2=3/2=1.5$. The power minus one is $3-1=2$, so the second piece is $2/2=1$. The deciding number is $1.5-1=0.5$.

Step 4. Read the box. The forgiving box is where the deciding number is below the regularity you are measuring the starting wave in. People usually measure the cubic-in-3D starting wave in the plain energy sense, which corresponds to regularity $1$. Since the deciding number $0.5$ is below $1$, this equation sits in the forgiving box: a short time window always produces a solution.

Step 5. Test a harder power for contrast. Keep $n=3$ but raise the power to $p=5$ (the "quintic" case). Now the power minus one is $4$, the second piece is $2/4=0.5$, and the deciding number is $1.5-0.5=1.0$. This equals the energy regularity $1$ exactly: the quintic equation in three dimensions sits in the balanced box, where only small starting waves are guaranteed a solution by this method. This balance point is famous — it is the energy-critical case.

What this tells us: a single arithmetic step, the deciding number $n/2-2/(p-1)$ against the regularity you start in, sorts the equation. The cubic in 3D is forgiving; the quintic in 3D sits exactly on the knife-edge; pushing the power higher would tip it into the box where this construction fails.

Check your understanding Beginner

Formal definition Intermediate+

Fix $n\ge 1$, a power $p>1$, and a sign $\mu \in \{+1,-1\}$ (defocusing/focusing). The semilinear Schrödinger equation (NLS) is $$i{\partial }_{t}u+\Delta u=\mu |u{|}^{p-1}u,u(0)=u_0,$$ for $u:I\times {\mathbb{R}}^n\to \mathbb{C}$ on a time interval $0\in I\subset \mathbb{R}$. The semilinear wave equation (NLW) is ${\partial }_{tt}u-\Delta u=\mu |u{|}^{p-1}u$ with data $(u(0),{\partial }_{t}u(0))=(u_0,u_1)$; the two run in parallel, the wave case carrying the half-derivative loss recorded in 02.21.01 and the first-order energy pair $(u_0,u_1)$ in place of the single datum $u_0$. The propagators are those of 02.21.02: $e^{it\Delta }$ for NLS and $(\cos (t\sqrt{-\Delta }),\sin (t\sqrt{-\Delta })/\sqrt{-\Delta })$ for NLW.

Definition (Duhamel / mild solution). A mild solution on $I$ is a function $u\in C(I;H^s)$ satisfying the Duhamel (variation-of-constants) formula $$u(t)=e^{it\Delta }{u}_0-i\mu {\int }_0^t{e}^{i(t-s)\Delta }(|u{|}^{p-1}u)(s)ds=:\Phi (u)(t).$$ A mild solution is a fixed point of the Duhamel map $\Phi$. The notion replaces classical differentiability by an integral identity that makes sense for low-regularity data and is equivalent to the PDE for smooth solutions.

Definition (scaling-critical Sobolev exponent). The free-equation scaling $u(t,x)\mapsto u_{\lambda }(t,x)={\lambda }^{2/(p-1)}u({\lambda }^{2}t,\lambda x)$ maps solutions of NLS to solutions. The homogeneous Sobolev norm $\Vert u_{\lambda }(0){\Vert }_{{\dot{H}}^s}$ is invariant under this scaling precisely at $$s_c=\frac{n}{2}-\frac{2}{p-1},$$ the scaling-critical (or critical) Sobolev exponent. A regularity $s$ is subcritical if $s>s_c$, critical if $s=s_c$, supercritical if $s<s_c$. The exponent ${\dot{H}}^{s_c}$ is the unique homogeneous Sobolev space at the same dilation weight as the equation. For NLW the critical exponent is the same $s_c$, computed against the wave scaling $u(t,x)\mapsto {\lambda }^{2/(p-1)}u(\lambda t,\lambda x)$.

Definition (Strichartz / solution space). Fix an admissible pair $(q,r)$ of 02.21.02. The Strichartz solution space is the mixed-norm space $X=C(I;H_x^s)\cap L_t^q(I;W_x^{s,r})$ — continuous in time into $H^s$, and $L^q$-in-time into the Sobolev space $W^{s,r}$ of 02.16.01 — with norm $\Vert u{\Vert }_X=\Vert u{\Vert }_{L_t^{\infty }{H}_x^s}+\Vert u{\Vert }_{L_t^q{W}_x^{s,r}}$. Local well-posedness is solved in $X$: the contraction runs on a closed ball of $X$ over $I$.

Definition (Hadamard well-posedness). The Cauchy problem is locally well-posed in $H^s$ if for each $u_0\in H^s$ there exist $T=T(\Vert u_0{\Vert }_{H^s})>0$, a unique mild solution $u\in X$ on $[-T,T]$, and the data-to-solution map $u_0\mapsto u$ is (locally Lipschitz) continuous from $H^s$ to $X$ — the existence, uniqueness, and continuous-dependence triple. Persistence of regularity adds that $u_0\in H^{s^{\prime }}$ with $s^{\prime }>s$ forces $u\in C([-T,T];H^{s^{\prime }})$ on the same interval.

Counterexamples to common slips Intermediate+

• Critical means the time interval stops helping. In the subcritical regime $s>s_c$ the smallness that closes the contraction comes from shrinking $T$: a positive power of $T$ multiplies the nonlinear term. At $s=s_c$ that power is exactly zero — the Strichartz norm of the linear evolution on the critical scale carries no factor of $T$ — so one must instead exploit smallness of $\Vert e^{it\Delta }{u}_0\Vert$ in the critical Strichartz norm, which holds for $u_0$ small or, for arbitrary $u_0$, on a short enough interval by dominated convergence. Treating the critical case as "subcritical with $s=s_c$" and hunting for a $T$-power is the standard error.

• Local well-posedness time depends on the norm, not just on the data. The existence time $T$ produced by the subcritical argument depends on $\Vert u_0{\Vert }_{H^s}$, not on finer features of $u_0$. Two data of equal norm get the same guaranteed $T$. This is why the blowup criterion is stated as: the solution extends as long as $\Vert u(t){\Vert }_{H^s}$ stays bounded.

• The nonlinearity must be Sobolev-admissible. Closing the contraction needs $|u{|}^{p-1}u$ to map $W^{s,r}$ to a dual Strichartz space, which requires the fractional-Leibniz/Moser estimate and a chain rule for $D^s$ of a $C^1$ (in fact $C^{\lceil s\rceil }$, or sufficiently smooth) nonlinearity. For non-integer $p$ the map $z\mapsto |z{|}^{p-1}z$ is only $C^{1,p-1}$ at the origin, capping the regularity $s$ for which the fractional chain rule applies; pushing $s$ past that cap is not a Strichartz failure but a nonlinearity-smoothness failure.

• Subcritical does not mean global. Local well-posedness with $T=T(\Vert u_0{\Vert }_{H^s})$ gives existence only up to the first time the norm might blow up. Globality requires an a-priori bound (conservation of mass/energy, or a Morawetz estimate), which is a separate theory. A solution can be locally well-posed and still blow up in finite time (focusing $L^2$-critical and energy-critical NLS).

Key theorem with proof Intermediate+

Theorem (subcritical local well-posedness for NLS; Ginibre-Velo 1979, Kato 1987). Let $n\ge 1$, $p>1$, $s\ge 0$ with $s>s_c=n/2-2/(p-1)$, and assume the nonlinearity $|u{|}^{p-1}u$ admits the fractional-Leibniz estimate at regularity $s$ (e.g. $p$ an odd integer, or $s<p$). Then for every $u_0\in H^s({\mathbb{R}}^n)$ there is $T=T(\Vert u_0{\Vert }_{H^s})>0$ and a unique mild solution $u\in C([-T,T];H^s)\cap L_t^q([-T,T];W_x^{s,r})$ for an admissible pair $(q,r)$, depending Lipschitz-continuously on $u_0$.

Proof. Choose an admissible Strichartz pair $(q,r)$ adapted to the nonlinearity, so that Hölder in space places $|u{|}^{p-1}u\in L_t^{q^{\prime }}{W}_x^{s,r^{\prime }}$ whenever $u\in L_t^q{W}_x^{s,r}$, with a power of $T$ to spare; the standing assumption $s>s_c$ is exactly the slack that produces a positive power $T^{\theta }$, $\theta >0$, after Hölder in time. Work on the complete metric space $B=\{u\in X_T:\Vert u{\Vert }_{X_T}\le 2M\}$ with $M=C\Vert u_0{\Vert }_{H^s}$, $X_T=C([-T,T];H^s)\cap L_t^q{W}_x^{s,r}$, metric $d(u,v)=\Vert u-v{\Vert }_{L_t^q{L}_x^r}$ (a complete metric on the closed ball; the high-regularity component is recovered by weak compactness).

Step 1 (self-map). For $u\in B$, the homogeneous Strichartz estimate of 02.21.02 gives $\Vert e^{it\Delta }{u}_0{\Vert }_{X_T}\le C\Vert u_0{\Vert }_{H^s}=M$. The retarded Strichartz estimate of 02.21.02 applied to the Duhamel term gives $$\Vert {\int }_0^t{e}^{i(t-s)\Delta }(|u{|}^{p-1}u)ds{\Vert }_{X_T}\le C\Vert |u{|}^{p-1}u{\Vert }_{L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}}.$$ By the fractional-Leibniz (Moser) estimate $\Vert |u{|}^{p-1}u{\Vert }_{W_x^{s,{\tilde{r}}^{\prime }}}\lesssim \Vert u{\Vert }_{L_x^a}^{p-1}\Vert u{\Vert }_{W_x^{s,r}}$ and Hölder in time with the leftover exponent, the right side is $\le C{T}^{\theta }\Vert u{\Vert }_{X_T}^p\le C{T}^{\theta }(2M{)}^p$. Hence $\Vert \Phi (u){\Vert }_{X_T}\le M+C{T}^{\theta }(2M{)}^p$. Choosing $T$ so small that $C{T}^{\theta }(2M{)}^p\le M$ gives $\Vert \Phi (u){\Vert }_{X_T}\le 2M$, so $\Phi :B\to B$.

Step 2 (contraction). For $u,v\in B$, the difference of nonlinearities obeys the pointwise bound $||u{|}^{p-1}u-|v{|}^{p-1}v|\lesssim (|u{|}^{p-1}+|v{|}^{p-1})|u-v|$. The retarded estimate and Hölder then give $$d(\Phi (u),\Phi (v))\le C{T}^{\theta }(\Vert u{\Vert }_{X_T}^{p-1}+\Vert v{\Vert }_{X_T}^{p-1})d(u,v)\le C{T}^{\theta }2(2M{)}^{p-1}d(u,v).$$ Shrinking $T$ further so that $2C{T}^{\theta }(2M{)}^{p-1}\le \frac{1}{2}$ makes $\Phi$ a contraction with constant $\frac{1}{2}$ on $(B,d)$.

Step 3 (fixed point). $B$ is closed in the complete metric $d$, so the Banach fixed-point theorem yields a unique fixed point $u\in B$, $\Phi (u)=u$. This $u$ is the mild solution. Uniqueness in the full space follows from the same contraction estimate run on any two solutions on a possibly shorter interval and a continuity/continuation argument. Lipschitz dependence: for data $u_0,v_0$, subtract the Duhamel formulas and run Step 2's estimate to get $\Vert u-v{\Vert }_{X_T}\le C\Vert u_0-v_0{\Vert }_{H^s}+\frac{1}{2}\Vert u-v{\Vert }_{X_T}$, hence $\Vert u-v{\Vert }_{X_T}\le 2C\Vert u_0-v_0{\Vert }_{H^s}$. $\square$

Bridge. This contraction is the foundational reason the Strichartz estimates of 02.21.02 were proved: it builds toward the global theory of nonlinear dispersive equations, where this local solution is iterated and extended, and it appears again in the energy-critical and mass-critical analysis where the same map is run in the critical norm. The central insight is that the only nonlinear input beyond Strichartz is one product estimate; this is exactly the mechanism by which a linear space-time bound becomes a nonlinear existence theorem. The subcritical $T^{\theta }$ slack generalises to the critical case by trading the time factor for smallness of the critical Strichartz norm, and putting these together the trichotomy $s\gtrless s_c$ is dual to the scaling weight of the equation: the bridge is the fixed-point set-up of 02.01.05, whose completeness is the abstract fact that the Picard iterates of the Duhamel map converge.

Exercises Intermediate+

Advanced results Master

Theorem (critical small-data well-posedness and scattering; Cazenave-Weissler 1990). Let $s=s_c=n/2-2/(p-1)\ge 0$ and let $(q,r)$ be the admissible pair with $q=r=p+1$-type scaling adapted to the critical ${\dot{H}}^{s_c}$ regularity. There is $\epsilon >0$ such that for every $u_0\in {\dot{H}}^{s_c}$ with $\Vert e^{it\Delta }{u}_0{\Vert }_{L_t^q{\dot{W}}_x^{s_c,r}(\mathbb{R})}\le \epsilon$ — in particular for all $u_0$ with $\Vert u_0{\Vert }_{{\dot{H}}^{s_c}}$ small — there is a global mild solution $u\in C(\mathbb{R};{\dot{H}}^{s_c})\cap L_t^q{\dot{W}}_x^{s_c,r}$, unique in that space, depending Lipschitz-continuously on $u_0$, and scattering: there exist $u_{\pm }\in {\dot{H}}^{s_c}$ with $\Vert u(t)-e^{it\Delta }{u}_{\pm }{\Vert }_{{\dot{H}}^{s_c}}\to 0$ as $t\to \pm \infty$ ^{[Cazenave-Weissler 1990]}. For arbitrary (not small) $u_0\in {\dot{H}}^{s_c}$, the same fixed point runs on a short interval where $\Vert e^{it\Delta }{u}_0{\Vert }_{L_t^q{\dot{W}}_x^{s_c,r}(I)}\le \epsilon$ by dominated convergence, giving local well-posedness without a smallness assumption on the data.

Theorem (blowup criterion / continuation). Let $u$ be the maximal-lifespan subcritical $H^s$ solution on $(-T_{-},T_{+})$. If $T_{+}<\infty$ then ${\lim }_{t\to T_{+}}\Vert u(t){\Vert }_{H^s}=\infty$; equivalently, the solution extends past any time at which the $H^s$ norm stays bounded. In the critical case the analogous criterion is that the solution extends as long as the critical Strichartz norm $\Vert u{\Vert }_{L_t^q{\dot{W}}_x^{s_c,r}}$ remains finite — a strictly stronger statement, since a bounded ${\dot{H}}^{s_c}$ norm does not by itself prevent blowup of the Strichartz norm. The Strichartz-norm criterion is the gateway to the induction-on-energy and concentration-compactness global theory.

Theorem (sharpness: supercritical ill-posedness). For $s<s_c$ the NLS Cauchy problem is ill-posed in $H^s$: the data-to-solution map fails to be uniformly continuous (indeed fails to be continuous) on bounded sets, by a scaling/modulation argument concentrating data at high frequency where the scaling pumps the $H^s$ norm down while the solution develops at unit scale. Thus the trichotomy is sharp: $s>s_c$ gives the contraction, $s=s_c$ gives small-data/short-time well-posedness, $s<s_c$ gives ill-posedness — the regularity threshold $s_c$ is the exact dividing line for this method and, for many ranges, for the equation itself ^{[Cazenave-Weissler 1990]}.

Theorem (NLW parallel and the half-derivative shift). For NLW ${\partial }_{tt}u-\Delta u=\mu |u{|}^{p-1}u$ the identical scheme runs with the wave propagators and the wave-admissible Strichartz pairs of 02.21.02: the energy space is ${\dot{H}}^s\times {\dot{H}}^{s-1}$ for the pair $(u,{\partial }_{t}u)$, the critical exponent is the same $s_c=n/2-2/(p-1)$, and the only structural change is that the wave dispersive exponent $(n-1)/2$ of 02.21.01 (versus $n/2$ for Schrödinger) shifts the admissible region, costing wave Strichartz estimates a derivative relative to Schrödinger. The contraction, persistence, blowup criterion, and continuous dependence transfer verbatim with $H^s$ replaced by the energy pair.

Synthesis. The contraction-mapping scheme is the central insight that unifies this unit: existence, uniqueness, and continuous dependence are not three theorems but one fixed point read three ways, and the foundational reason all three hold is the single retarded Strichartz estimate of 02.21.02 fed into one fractional-Leibniz product estimate. Putting these together, the scaling exponent $s_c=n/2-2/(p-1)$ is exactly the dilation weight at which the equation is invariant, and the subcritical/critical/supercritical trichotomy is dual to whether the time-interval factor $T^{\theta }$ is positive, zero, or absent — this is exactly the same arithmetic that, in 02.21.02, became the admissibility line $2/q+n/r=n/2$, so the Strichartz scaling and the well-posedness scaling are one relation. The critical case generalises the subcritical one by trading the vanishing $T^{\theta }$ for smallness of the critical Strichartz norm, which appears again in the global concentration-compactness program; what looks from a distance like three separate phenomena — Strichartz integrability, scaling criticality, and Picard iteration — is one curved-dispersion mechanism, and the bridge to the global theory is the blowup criterion, which converts a Strichartz-norm bound into unconditional global existence.

Full proof set Master

Proposition 1 (the Duhamel map is well-defined into the Strichartz space). Let $(q,r)$ be admissible and $s\ge 0$. If $u_0\in H^s$ and $F\in L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}([-T,T])$ for an admissible $(\tilde{q},\tilde{r})$, then ${\Phi }_0(t)=e^{it\Delta }{u}_0-i\mu {\int }_0^t{e}^{i(t-s^{\prime })\Delta }F(s^{\prime })d{s}^{\prime }$ lies in $C([-T,T];H^s)\cap L_t^q{W}_x^{s,r}$ with $\Vert {\Phi }_0{\Vert }_{X_T}\le C(\Vert u_0{\Vert }_{H^s}+\Vert F{\Vert }_{L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}})$.

Proof. Linearity splits ${\Phi }_0$ into the free evolution and the Duhamel term. The homogeneous Strichartz estimate of 02.21.02, applied to $⟨\nabla {⟩}^s$ of the datum (commuting the Bessel potential through the propagator, which is a Fourier multiplier), gives $\Vert e^{it\Delta }{u}_0{\Vert }_{L_t^q{W}_x^{s,r}}\lesssim \Vert u_0{\Vert }_{H^s}$ and $\Vert e^{it\Delta }{u}_0{\Vert }_{L_t^{\infty }{H}_x^s}=\Vert u_0{\Vert }_{H^s}$ by unitarity. The retarded estimate of 02.21.02, likewise applied to $⟨\nabla {⟩}^{s}F$, gives $\Vert {\int }_0^t{e}^{i(t-s^{\prime })\Delta }Fd{s}^{\prime }{\Vert }_{L_t^q{W}_x^{s,r}\cap L_t^{\infty }{H}_x^s}\lesssim \Vert F{\Vert }_{L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}}$. Continuity in time into $H^s$ is the strong continuity of the unitary group plus dominated convergence on the integral. Summing yields the stated bound. $\square$

Proposition 2 (fractional-Leibniz / Moser product estimate, statement and use). Let $s>0$, $1<r,r_1,r_2,a<\infty$ with $1/r^{\prime }=(p-1)/a+1/r_2$ and the Sobolev embedding $W^{s,r}\hookrightarrow L^a$ available. Then $\Vert |u{|}^{p-1}u{\Vert }_{W^{s,r^{\prime }}}\lesssim \Vert u{\Vert }_{L^a}^{p-1}\Vert u{\Vert }_{W^{s,r}}$ whenever $z\mapsto |z{|}^{p-1}z$ is sufficiently smooth ($p$ odd integer, or $s\le p$).

Proof. By the Kato-Ponce fractional Leibniz rule, for $F(u)=|u{|}^{p-1}u$, $\Vert D^{s}F(u){\Vert }_{L^{r^{\prime }}}\lesssim \Vert F^{\prime }(u){\Vert }_{L^b}\Vert D^{s}u{\Vert }_{L^r}$ with $1/r^{\prime }=1/b+1/r$, together with the fractional chain rule $\Vert D^{s}F(u){\Vert }_{L^{r^{\prime }}}\lesssim \Vert F^{\prime }(u){\Vert }_{L^b}\Vert D^{s}u{\Vert }_{L^r}$ valid for $F\in C^{\lceil s\rceil }$ (or $C^{1,p-1}$ when $0<s\le 1$). Since $|F^{\prime }(u)|\lesssim |u{|}^{p-1}$, $\Vert F^{\prime }(u){\Vert }_{L^b}\lesssim \Vert u{\Vert }_{L^{(p-1)b}}^{p-1}$. Setting $a=(p-1)b$ recovers the stated Hölder relation $1/r^{\prime }=(p-1)/a+1/r$ and the bound $\Vert u{\Vert }_{L^a}^{p-1}\Vert D^{s}u{\Vert }_{L^r}$; adding the $L^{r^{\prime }}$ bound of the function itself (the $0$-derivative term) completes the $W^{s,r^{\prime }}$ estimate. The smoothness hypothesis is exactly what licenses the fractional chain rule at order $s$. $\square$

Proposition 3 (Banach fixed point on the Strichartz ball). Let $(B,d)$ be the closed ball of radius $2M$ in $X_T$ with the metric $d(u,v)=\Vert u-v{\Vert }_{L_t^q{L}_x^r}$. If $\Phi$ maps $B$ to $B$ and satisfies $d(\Phi u,\Phi v)\le \kappa d(u,v)$ with $\kappa <1$, then $\Phi$ has a unique fixed point in $B$.

Proof. $(L_t^q{L}_x^r,d)$ is complete and $B$ is closed and bounded in $X_T\subset L_t^q{L}_x^r$, hence $d$-complete (a $d$-Cauchy sequence in $B$ converges in $L_t^q{L}_x^r$, and the uniform $X_T$ bound passes to the limit by weak-* lower semicontinuity of the $X_T$ norm, keeping the limit in $B$). The Banach contraction principle of 02.01.05 applied to the complete metric space $(B,d)$ and the $\kappa$-Lipschitz self-map $\Phi$ yields a unique fixed point, obtained as the limit of the Picard iterates $u_{k+1}=\Phi (u_k)$ from any $u_0\in B$, with the geometric error bound $d(u_k,u_{\infty })\le {\kappa }^k(1-\kappa {)}^{-1}d(u_1,u_0)$. $\square$

Proposition 4 (continuous dependence with explicit modulus). Under the subcritical hypotheses, the solution map $u_0\mapsto u$ is Lipschitz from a ball in $H^s$ to $X_T$, with $\Vert u-v{\Vert }_{X_T}\le 2C\Vert u_0-v_0{\Vert }_{H^s}$.

Proof. Let $u,v$ solve with data $u_0,v_0$. Subtract the Duhamel formulas: $u-v=e^{it\Delta }(u_0-v_0)-i\mu {\int }_0^t{e}^{i(t-s^{\prime })\Delta }(F(u)-F(v))d{s}^{\prime }$. Proposition 1 bounds the free term by $C\Vert u_0-v_0{\Vert }_{H^s}$. Proposition 2 plus the pointwise difference bound of Exercise 5 gives $\Vert F(u)-F(v){\Vert }_{L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}}\le C{T}^{\theta }(\Vert u{\Vert }_{X_T}^{p-1}+\Vert v{\Vert }_{X_T}^{p-1})\Vert u-v{\Vert }_{X_T}\le \frac{1}{2}\Vert u-v{\Vert }_{X_T}$ after the contraction choice of $T$. Hence $\Vert u-v{\Vert }_{X_T}\le C\Vert u_0-v_0{\Vert }_{H^s}+\frac{1}{2}\Vert u-v{\Vert }_{X_T}$, and absorbing the last term gives $\Vert u-v{\Vert }_{X_T}\le 2C\Vert u_0-v_0{\Vert }_{H^s}$. $\square$

Proposition 5 (vanishing of the time factor at criticality). In the diagonal subcritical set-up of Exercise 6 the surplus time exponent is $\theta =1/{\tilde{q}}^{\prime }-p/q$, and $\theta =0$ exactly when $s=s_c$.

Proof. The nonlinearity $|u{|}^{p-1}u$ with $u\in L_t^q{L}_x^r$ has space-time integrability $L_t^{q/p}{L}_x^{r/p}$. The retarded estimate requires the input in $L_t^{{\tilde{q}}^{\prime }}{L}_x^{{\tilde{r}}^{\prime }}$ with ${\tilde{r}}^{\prime }=r/p$, fixing $\tilde{q}$ admissibly. On $[0,T]$, Hölder in time against $1_{[0,T]}$ between $L_t^{q/p}$ and $L_t^{{\tilde{q}}^{\prime }}$ costs $T^{\theta }$ with $\theta =1/{\tilde{q}}^{\prime }-p/q$ provided $\theta \ge 0$. A dimensional-analysis/scaling check: the full Duhamel estimate is scale-invariant precisely at $s=s_c$, where no power of $T$ (the one dimensionful quantity) can appear; hence $\theta =0⟺s=s_c$, $\theta >0⟺s>s_c$, and for $s<s_c$ the exponent would be negative, signalling that no positive power of $T$ helps — the supercritical obstruction. $\square$

Proposition 6 (uniqueness in the energy/Strichartz class). Two mild solutions $u,v\in C([-T,T];H^s)\cap L_t^q{W}_x^{s,r}$ with the same data agree on $[-T,T]$.

Proof. On a possibly smaller interval $[0,\tau ]$ the contraction estimate of Step 2 gives $\Vert u-v{\Vert }_{X_{\tau }}\le \kappa (\tau )\Vert u-v{\Vert }_{X_{\tau }}$ with $\kappa (\tau )=C{T}^{\theta }(\dots )\to 0$ as $\tau \to 0$; choosing $\tau$ with $\kappa (\tau )<1$ forces $u=v$ on $[0,\tau ]$. The set $\{t:u=v\text{ on }[0,t]\}$ is then open and closed in $[0,T]$ and nonempty, hence all of $[0,T]$; the same on $[-T,0]$. $\square$

Connections Master

• The entire linear input is the homogeneous and retarded Strichartz estimates of 02.21.02: the homogeneous estimate controls $e^{it\Delta }{u}_0$ and the retarded estimate controls the Duhamel term, so the contraction is, structurally, the nonlinear shadow of the $T{T}^{\ast }$ method, and the admissibility line $2/q+n/r=n/2$ of that unit reappears here as the criterion that makes the work space close under the nonlinearity.

• The nonlinear product estimate that closes the contraction is the fractional-Leibniz/Moser estimate built on the Sobolev embeddings $W^{s,r}\hookrightarrow L^a$ of 02.16.01; without the Gagliardo-Nirenberg-Sobolev inequalities of that unit there is no way to convert $L_t^q{W}_x^{s,r}$ control of $u$ into $L_t^{{\tilde{q}}^{\prime }}{W}_x^{s,{\tilde{r}}^{\prime }}$ control of $|u{|}^{p-1}u$, so the entire nonlinear half of the argument is downstream of 02.16.01.

• The fixed point is the Banach contraction principle on the complete metric ball of 02.01.05: existence, uniqueness, and Lipschitz dependence are the three outputs of one contraction, and the completeness that makes the Picard iterates converge is exactly the metric-completeness developed in that unit, so the well-posedness package is the dispersive instance of the abstract iteration scheme.

Historical & philosophical context Master

The contraction-mapping treatment of nonlinear Schrödinger equations was set out systematically by Jean Ginibre and Giorgio Velo, whose 1979 Journal of Functional Analysis paper On a class of nonlinear Schrödinger equations ^{[Ginibre-Velo 1979]} established local and global existence in the energy space by combining the linear estimates with energy conservation, working before the Strichartz machinery was in its modern form. Tosio Kato's 1987 Annales de l'IHP paper On nonlinear Schrödinger equations ^{[Kato 1987]} recast the local theory in the $H^s$ Sobolev scale and made the role of the Strichartz estimates explicit, isolating the fractional-derivative product estimates as the nonlinear input and giving the persistence-of-regularity and uniqueness statements in the form used here.

The critical theory — small-data global existence and the identification of $s_c=n/2-2/(p-1)$ as the sharp scaling threshold — is due to Thierry Cazenave and Fred Weissler, whose 1990 Nonlinear Analysis paper The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$ ^{[Cazenave-Weissler 1990]} ran the fixed point in the critical Strichartz norm and proved the global well-posedness, scattering, and uniqueness package at $s=s_c$, replacing the time-interval smallness of the subcritical theory by smallness of the critical norm. The fixed-point principle itself is Stefan Banach's, from his 1922 Fundamenta Mathematicae thesis ^{[Banach 1922]}; its appearance as the engine of nonlinear dispersive well-posedness, with the Strichartz space as the complete metric space, is the synthesis these later authors supplied. Terence Tao's 2006 lecture notes organize the subcritical, critical, and supercritical regimes into the trichotomy presented here.

Bibliography Master

@article{GinibreVelo1979,
  author  = {Ginibre, Jean and Velo, Giorgio},
  title   = {On a class of nonlinear {S}chr\"odinger equations},
  journal = {Journal of Functional Analysis},
  volume  = {32},
  year    = {1979},
  pages   = {1--71}
}

@article{CazenaveWeissler1990,
  author  = {Cazenave, Thierry and Weissler, Fred B.},
  title   = {The {C}auchy problem for the critical nonlinear {S}chr\"odinger equation in {$H^s$}},
  journal = {Nonlinear Analysis: Theory, Methods \& Applications},
  volume  = {14},
  year    = {1990},
  pages   = {807--836}
}

@article{Kato1987,
  author  = {Kato, Tosio},
  title   = {On nonlinear {S}chr\"odinger equations},
  journal = {Annales de l'Institut Henri Poincar\'e, Physique Th\'eorique},
  volume  = {46},
  year    = {1987},
  pages   = {113--129}
}

@book{Cazenave2003,
  author    = {Cazenave, Thierry},
  title     = {Semilinear {S}chr\"odinger Equations},
  series    = {Courant Lecture Notes in Mathematics},
  volume    = {10},
  publisher = {American Mathematical Society},
  year      = {2003}
}

@article{Banach1922,
  author  = {Banach, Stefan},
  title   = {Sur les op\'erations dans les ensembles abstraits et leur application aux \'equations int\'egrales},
  journal = {Fundamenta Mathematicae},
  volume  = {3},
  year    = {1922},
  pages   = {133--181}
}

@book{Tao2006,
  author    = {Tao, Terence},
  title     = {Nonlinear Dispersive Equations: Local and Global Analysis},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {106},
  publisher = {American Mathematical Society},
  year      = {2006}
}