27

On the effects of spatial expansion and contraction
on several semilinear partial differential equations
Makoto NAKAMURA( 中村 誠

(山形大学

理)) *

Abstract

The derivation of several second order partial differentiaı equations is con‐sidered based on the scalar‐field equation and its non‐relativistic limit in theuniform and isotropic space. The field equation is derived as the Euler‐Lagrange
equation for a Lagrangian given in the spacetime which is a solution of the Ein‐
stein equation for non‐Hermitian line elements. Some results on the Cauchy
problem of the limit equation are introduced. The derivation of some equa‐tions for vectors and their energy estimates are also introduced. A dissipative
property of the spatial expansion is remarked.

1

Introduction

In this paper, we first consider the derivation of several semilinear second order par‐tial differential equations based on the field equation and its nonrelativistic limit in
the spacetime generated by the Einstein equation for complex‐valued line elements.

Second, we consider the effect of the spatial variation (expansion or contraction) on
the Cauchy problem of the limit equation.
For m, c, \lambda, \hslash(\neq 0)\in \mathbb{R} and 1\leq p<\infty , let us consider the semilinear Klein‐
Gordon equation\partial_{t}^{2}\phi-c^{2}\triangle_{x}\phi+\frac{m^{2}c^{4} {h^{2} \phi-c^{2} \lambda|\phi|^{p-1}\phi=0 ,(1.1)

the semilinear Schrödinger equation\pm i\frac{2m}{\hslash}\partial_{t}u+\triangle_{x}u+\lambda|u|^{p-1}u=0 ,(1.2)

the semilinear elliptic equation\partial_{t}^{2}\phi+c^{2}\triangle_{x}\phi+\frac{m^{2}c^{4} {h^{2} \phi-c^{2} \lambda|\phi|^{p-1}\phi=0 ,(1.3)’Faculty of Science, Yamagata University, Kojirakawa‐machi 1‐4‐12, Yamagata 990‐8560,
JAPAN. E‐‐mail: nakamura@sci.kj. yamagata‐u. ac. jp

28and the semilinear parabolic equation\frac{2m}{\hslash}\partial_{t}u-\triangle.u-i\lambda|u|^{p-1}u=0 ,(1.4)

where we have put \triangle_{x} := \sum_{j=1}^{n}\partial^{2}/(\partial x^{j})^{2} . For the elliptic equation (1.3), thevariable ct can be naturally regarded as one of spatial variables. The terms \lambda|\phi|^{p-1}\phiand \lambda|u|^{p-1}u are fundamental semilinear terms in the nonlinear theory to describe
the self‐interaction of the solution. For the last parabolic equation (1.4), we notethat the dimension of \hslash/m in the SI units is M^{2}S^{-1} ( M : meter, S : second), whichis equivalent to the dimension of the thermal diffusivity K_{1} of the heat equation\partial_{t}u-K_{1}\triangle_{x}u=0 , and also to the dimension of the diffusion coefficient K_{2} of the
diffusion equation \partial_{t}u-K_{2}\triangle_{x}u=0.
To consider the derivation of the above equations, let us consider the follow‐
ing line element. For any natural number n and any fixed real numbers \omega=(\omega^{0}, \cdots , \omega^{n})\in(-\pi/2, \pi/2]^{1+n} , we consider a(1+n) ‐dimensional space \mathbb{M}^{1+n} de‐fined by\mathbb{M}^{1+n} :=\{z\in \mathbb{C}^{1+n}|z^{\alpha}=x^{\alpha} e^{i\omega^{\alpha}}, x^{\alpha}\in \mathbb{R}, 0\leq\alpha\leq n\},where \mathb {C} denotes the set of complex numbers. We consider a generalization of theEinstein equation for non‐Hermitian complex line elements of the form g_{\alpha\beta}(z)dz^{\alpha}dz^{\beta},
where \{g_{\alpha\beta}\}_{0\leq\alpha,\beta\leq n} are complex‐valued functions for z= (z^{0} . , z^{n})\in \mathbb{M}^{1+n}.
Under the cosmological principle, we give the solution of the generalized Einsteinequation asg_{\alpha\beta}dz^{\alpha}dz^{\beta}=-c^{2}(dz^{0})^{2}+a(z^{0})^{2}q^{2}(1+ \frac{k^{2}r^{2} {4})^{-2}\sum_{j=1}^{n}(dz^{j})^{2}
where c>0 is the speed of light, q(\neq 0), k\in \mathbb{C} are constants, r

,

(1.5)

:= \{\sum_{\alpha=1}^{n}(z^{\alpha})^{2}\}^{1/2}and a(\cdot) is a complex‐valued function which denotes the scale‐function of the space.
There is a large body of literature on the generalization of the Einstein equation for

Hermitian line elements and general dimensions (see e.g. [1, 2, 3, 4, 5, 6]).
For any function f on \mathbb{M}^{1+n} , we consider the derivative \partial_{\alpha}f(z) for z\in \mathbb{M}^{1+n} by\partial_{\alpha}f(z) := \lim_{har ow 0} \frac{f(z^{0},\cdots,z^{\alpha- {\imath} ,z^{\alpha}+he^{i\omega^{\alpha} ,z^{\alpha+1},\cdot\cdot,z^{n})-f(z)} {he^{i\omega^{\alpha} } .

(1.6)

h\in \mathbb{R}\backslash \{0\}

Since

z=

(z^{0} , z^{n})\in \mathbb{M}^{{\imath}+n} is parametrized by x=(x^{0} . , x^{n})\in \mathbb{R}^{1+n} by the

relationz^{\alpha}=x^{\alpha}e^{i\omega^{\alpha}}(1.7)

if we put f_{*}(x) :=f(z) with (1.7), then we have \partial_{\alpha}f(z)=e^{-i\omega^{\alpha}}\partial f_{*}(x)/\partial x^{\alpha} . Let us
consider the background spacetime \mathbb{M}^{1+n} with the line element (1.5). We put q=1
and k=0 in (1.5). As the equation of motion of the massive scalar field describedby a complex‐valued function \phi=\phi (z^{0}, \cdots , z^{n}) with the mass m and a potential

29\lambda|\phi|^{p-1}\phi^{2}/(p+1) for \lambda\in \mathbb{C} and 1\leq p<\infty . we derive the second order differential
equation‐\frac{1}{c^{2} (\partial_{0}^{2}+\frac{n\partial_{0}a}{a}\partial_{0}+\frac{m^ {2}c^{4} {\hslash^{2} )\phi+\frac{1}{a^{2} \triangle_{z}\phi+\lambda|\phi|^{p-1} \phi=0 ,(1.8)

where \hslash is the Planck constant, \partial_{0} :=\partial/\partial z^{0} and \triangle_{z} := \sum_{j=1}^{n}\partial^{2}/(\partial z^{j})^{2} . We also

show that the nonrelativistic limit of (1.8) yields the equation\pm i\frac{2m}{\hslash}\partial_{0}u+\frac{1}{a^{2} \triangle_{z}u+ \lambda|uw|^{p-1}u=0(z^{0}, \cdots , z^{n}) (see (2.18), below), whereis a weight function defined by w(z^{0}) :=b_{0}(a(0)/a(z^{0}))^{n/2} forwith a suitable transform from \phi to:=(-1)^{1/2} and

i

w

(1.9)

u=u

a constant b_{0}\in \mathbb{C} . By the transform (1.7), the equations (1.8) and (1.9) give theequations (1. 1), (1.2), (1.3) and (1.4) when a (.)ı.
=

2Derivation of the field equation

In this section, let us derive the line element (1.5). In the following, Greek letters\alpha, \beta, \gamma,\cdot\cdot\cdot

run from 0 to n , Latin letters j, k, l,\cdot\cdot\cdot

run from 1 to n . We use theEinstein rule for the sum of indices of tensors, for example, T_{\alpha}^{\alpha} := \sum_{\alpha=0}^{n}T_{\alpha}^{\alpha} andT^{j_{j}} := \sum_{j=1}^{n}T^{j_{j}} . For any function f on \mathbb{M}^{1+n} , we put f_{*}(x) :=f(z) with (1.7). We
define the integral \int_{\mathbb{M}^{1+n}}f(z)dz by\int_{\mathb {M}^{1+n} f(z)dz :=e^{i\Sigma_{\alpha=0}^{n}\omega^{\alpha} \int_ {\mathb {R}^{1+n} f_{*}(x)dx .

(2.1)

We consider a bilinear symmetric complex‐valued functional \langle\cdot, \cdot\rangle on the vector

space spanned by the vectors \{\partial_{\alpha}\}_{0\leq\alpha\leq n} . We put g_{\alpha\beta}(z) :=\langle\partial_{\alpha}, \partial_{\beta}\rangle . We denote by(g_{\alpha\beta}(z)) the (1+n)x(1+n) ‐matrix whose components are given by \{g_{\alpha\beta}(z)\}_{0\leq\alpha,\beta\leq n}.
Put g(z) :=\det(g_{\alpha\beta}(z)) . Let (g^{\alpha\beta}(z)) be the inverse matrix of (g_{\alpha\beta}(z)) . We consider
a line element-(cd\tau)^{2}=(dP)^{2} :=g_{\alpha\beta}(z)dz^{\alpha}dz^{\beta} ,(2.2)

where \tau denotes the proper time and we take the square root of (cd\tau)^{2} as\arg(cd\tau)\leq\pi . We define dz by-\pi<

dz=dz^{0} \wedge\cdots\wedge dz^{n}:=\sum_{\sigma}sgn(\sigma)dz^{\sigma(0)} \cdots dz^{\sigma(n)},where \sigma denotes the permutation of \{0, \cdot\cdot\cdot , n\}.

We define the Christoffel symbol by\Gam a_{\beta\gam a}^{\alpha}:=\frac{1}{2}g^{\alpha\delta}(\partial_{\beta}g_ {\delta\gam a}+\partial_{\gam a}g_{\beta\delta}-\partial_{\delta}g_{\beta\gam a} ) .
We define the covariant derivative \nabla_{\beta} for T^{\alpha} by\nabla_{\beta}T^{\alpha}(z) :=\partial_{\beta}T^{\alpha}(z)+ \Gamma_{\beta\gamma}^{\alpha}(z)\mathcal{I}^{\eta}(z)

.

(2.3)

30
In general, we define\nabla_{\delta\mu\mu\cdots\mu\nu}T\nu\cdots:=\partial_{\delta v}T+ \Gamma_{\delta\varepsilon}^{\alpha}T+\Gamma T^{\alpha\in}\ldots\ldots+\cdots-\Gamma^{\varepsilon}\mu\delta\nu\mu\varepsilon
for any tensor T^{\alpha\beta}\ldots\mu\nu\cdots
We define the Riemann curvature tensor

R_{\alpha\beta\gamma}^{\delta}:=\partial_{\beta}\Gamma_{\alpha\gamma}^{\delta}- \partial_{\gamma}\Gamma_{\alpha\beta}^{\delta}+\Gamma_{\varepsilon\beta} ^{\delta}\Gamma_{\alpha\gamma}^{\varepsilon}-\Gamma_{\varepsilon\gamma}^{\delta} \Gamma_{\alpha\beta}^{\varepsilon}
which is derived from R_{\alpha\beta\gamma}^{\delta}T^{\alpha}=(\nabla_{\beta}\nabla_{\gamma}-\nabla_ {\gamma}\nabla_{\beta})T^{\delta} . We define the Ricci tensor

R_{\alpha\beta} :=R^{\gamma_{\alpha\beta\gamma}} , and the scalar curvature R :=g^{\alpha\beta}R_{\alpha\beta} . We define the Einstein tensor
by G_{\alpha\beta} :=R_{\alpha\beta}-g_{\alpha\beta}R/2 . The change of upper and lower indices is done by g_{\alpha\beta}
and g^{\alpha\beta} , for example, G^{\alpha_{\beta} :=g^{\alpha\gamma}G_{\gamma\beta}.
Let \Lambda\in \mathbb{C} be a constant, which is called the cosmological constant. Let us con‐sider the variation by g_{\alpha\beta} of the Einstein‐Hilbert action \int_{\mathbb{M}^{1+n}}(R+2\Lambda)(-g)^{{\imath}/2}dz.
Then the Euler‐Lagrange equation for the action is given by the Einstein equation
G_{\alpha\beta}-\Lambda g_{\alpha\beta}=0 in the vacuum. For a stress‐energy tensor T^{\alpha_{\beta} , we define the(1+n) ‐dimensional Einstein equation

G_{\beta}^{\alpha}-\Lambda g_{\beta}^{\alpha}=\kappa T^{\alpha_{\beta} ,(2.4)

where \kappa is a constant and we assume that \kappa is written as \kappa=\kappa_{0}/c^{4} for some constant\kappa_{0} which is independent of c .

For the case n=3 and real line elements, the constantis called the Einstein gravitational constant which is given by \kappa=8\pi \mathcal{G}/c^{4} , where \mathcal{G}is the Newton gravitational constant. For the case n\geq 3 and complex line elements,
we are able to generalize the constant \kappa to\kappa\kap a=\frac{2(n-1)\pi^{n/2}\mathcal{G} {(n-2)\Gam a(n/2)c^{4}

,

(2.5)

where \Gamma denotes the gamma function. We have obtained the generalized Einsteinequation (2.4) with (2.5) for complex line elements.
Let us derive the line element (1.5) as the solution of the Einstein equation (2.4).
We assume that the space is uniform and isotropic, and we consider the line element

g_{\alpha\beta}dz^{\alpha}dz^{\beta} :=-c^{2}(dz^{0})^{2}+e^{h(z^{0}) e^{f(r)} \sum_{j=1}^{n}(dz^{j})^{2}

,

(2.6)

where h and f are complex‐valued functions. This line element is uniform inthe sense that for any two points P and Q in \mathbb{C}^{n} , the ratio of the coefficientse^{h(z^{0})}e^{f(r(P))}/e^{h(z^{0})}e^{f(r(Q))} is independent of z^{0}.
By direct calculations, we have G_{j}^{0}=G_{0}^{j}=0,

G_{0}^{0}:= \frac{n-1}{2c^{2} \{\frac{n}{4}(\partial_{0}h)^{2}-c^{2}e^{-h-f}(f" +(n-1)\frac{f'}{r}+\frac{n-2}{4}(f')^{2})\},

31
31
and

G^{j_{k} :=g^{j_{k} \{\frac{n-1}{2c^{2} (\partial_{0}^{2}h+\frac{n}{4} (\partial_{0}h)^{2})- \frac{n-2}{2}e^{-h-f}(f"+(n-2)\frac{f'}{r}+\frac{n-3}{4}(f')^{2})\}+ \frac{n-2}{2}e^{-h-f}(f"-\frac{f^{l} {r}-\frac{(f')^{2} {2})\frac{z^{j}z^{k} {r^{2} ,where f' :=df/dr . Since the space is isotropic, the coefficient of z^{j}z^{k} must vanish.
So that, f must satisfy f"-f'/r-(f')^{2}/2=0 , by which we obtaine^{f(r)}=q^{2}(1+ \frac{k^{2}r^{2} {4})^{-2}for constants(2.7)

q(\neq 0), k\in \mathbb{C} . We define a functiona(z^{0}) :=e^{h(z^{0})/2}(2.8)

Let us consider the stress‐energy tensor T_{\beta}^{\alpha} of the perfect fluid

T_{\beta}^{\alpha} := diag (\rho c^{2}, -p, \cdots , -p)
for constant density \rho and pressure p . We put \overline{\rho}:=\rho+\Lambda/KC^{2} and \overline{p}:=p-\Lambda/\kappa.
Then (2.4) is rewritten as G_{\beta}^{\alpha}=\kappa\cdot diag(\overline{\rho}c^{2}, -\overline{p}, \cdots , -\tilde{p}) . This equation showsthat the cosmological constant \Lambda>0 is regarded as the energy which has positive

density and negative pressure in the vacuum \rho=p=0 for \kappa>0 , by which we regard
the cosmological constant \Lambda as “the dark energy The equation G^{0_{0} =\kappa\overline{\rho}c^{2}g^{0_{0} isrewritten as\frac{n-1}{2}\{(\frac{\partial_{0}a {ca})^{2}+\frac{k^{2} {q^{2}a^{2} \}=\frac {\kap ac^{2} {n}\cdot\overline{\rho}

.

(2.9)

The equation G_{k}^{j}=-\kappa\overline{p}g^{j_{k}} is rewritten as\frac{n-1}{2}\{\frac{2}{n-2} . \frac{\partial_{0}^{2}a {c^{2}a + (\frac{\partial_{0}a {ca})^{2}+\frac{k^{2} {q^{2}a^{2} \}=-\frac{\kap a}{n-2}. \overline{p},(2.10)

which is rewritten as the Raychaudhuri equation\frac{\partial_{0}^{2}a {c^{2}a =-\frac{n-2}{n-1} \cdot\kap a(\frac{\overline{\rho}c^{2} {n}+\frac{\overline{p} {\bulet-2})(2.11)by (2.9). Multiplying a^{n} to the both sides in (2.9), taking the derivative by z^{0}variable, and using (2.10), we have the conservation of the mass\partial_{0}(\overline{\rho}c^{2}a^{n})+\overline{p}\partial_{0}a^{n}=0 .

(2.12)

32
For any number \sigma , we assume the equation of state\overline{p}=\sigma\overline{\rho}c^{2}(2.13)

Then a(z^{0}) must satisfy\frac{\partial_{0}^{2}a(z^{0}) {c^{2}a(z^{0}) =-\frac{n-2+n\sigma}{n(n-1)} \kap a\overline{\rho}c^{2}.

with\overline{\rho}=\frac{n-1}{2}\cdot\frac{n}{\kap a c^{4} \cdot(\frac{\partial_{0}a(0)}{a(0)} ^{2} (\frac{a(0)}{a(z^{0}) ^{n({\imath}+ \sigma)}(2.14)a(z^{0}):=\{ begin{ar y}{l a(0)1+\frac{n(1+\sigma)\partial_{0}a(0)z^{0}{2a(0)}^{2/n(1+\sigma)} if \sigma\neq-1, a(0)\exp(\frac{\partial_{0}a(0)z^{0}{a(0)} if\sigma=-1. \end{ar y}(2.15)by (2.11) and (2.12). We consider the solution which has the curvature k=0 givenby

Let us derive the equations (1.8) and (1.9). For any \lambda\in \mathbb{C} and any complex‐
valued C^{2} function \phi on \mathbb{M}^{ \imath}+n} , we define the LagrangianL( \phi):=-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi- \frac{1}{2}(\frac{mc}{\hslash})^{2}\phi^{2}+\frac{\lambda}{p+1}|\phi|^{p-1}\phi^ {2}.
Then with the constraint condition \arg\delta\phi=\arg\phi , the Euler‐Lagrange equation for
the action \int_{\mathbb{M}^{1+n}}L(\phi)(-g)^{1/2}dz is given by\frac{1}{(-g)^{1/2} \partial_{a}( -g)^{1/2}g^{\alpha\beta}\partial_{\beta} \phi)-(\frac{mc}{\hslash})^{2}\phi+\lambda|\phi|^{p-1}\phi=0 ,(2.16)

which is rewritten as the equation (ı.8). For any constant b_{0}\in \mathbb{C} , we define a weightfunction w(z^{0}) and a function b(z^{0}) byw(z^{0}) :=b_{0}( \frac{a(0)}{a(z^{0})})^{n/2} b(z^{0}) :=w(z^{0})\exp(\mp i\frac{mc^{2} {\hslash}z^{0}),(2.17)

where we note b(0)=b_{0} . We t\cdot ransform \phi to u by the equation\phi (z^{0}, \cdots , z^{n})=u(z^{0}, \cdots , z^{n})b ( z 0).

(2.18)

We assume mz^{0}/\hslash\in \mathbb{R} . Then the nonrelativistic limit (carrow\infty) of this equation
yields (1.9).

3

The Cauchy problem

In this section, we introduce some results on the Cauchy problem of the equation(1.9) without proofs. The equation (1.9) is rewritten as\pm i\frac{2m}{\hslash}\partial_{t}u+\frac{1}{a^{2}e^{2i\omega} \triangle u- \frac{\lambda}{e^{2i\omega} |uW|^{p-1}u=0

(3.1)

33
by a transformation (1.7) with t :=x^{0}, \omega^{0}=0, \omega :=\omega^{1}==\omega^{n} .

We consider

how the spatial variance affects the existence of the solutions. Let T_{0} be the maximal

existence time of the scale‐function a(\cdot) defined by (2.15). Since the equation (3.1)
has a variable coefficient, we use a change of variable s=s(t) := \int_{0}^{t}a(\tau)^{-2}d\tau.
We put S_{0} :=s(T_{0}) . We use conventions a(\mathcal{S}) :=a(t(\mathcal{S})) and w(s) :=w(t(s)) fors\in[0, S_{0}) as far as there is no fear of confusion. A direct computation shows

S_{0}=\{ begin{ar ay}{l} \frac{2}{a_{0}a_{1}(4-n(1+\sigma) } ifa_{1}(4-n(1+\sigma) >0, \infty ifa_{1}(4-n(1+\sigma) \leq0. \end{ar ay}
For 0\leq\mu_{0}<n/2 and 0<S\leq S_{0} , we consider the Cauchy problem given by\{ begin{ar y}{l \pmi\frac{2m}{\hsla h}\partial_{s}u(s,x)+\frac{1}e^{2i\omega}\triangleu(s, x)-\frac{\lambda (s)^{2}{e^2i\omega}(|uw|^{p-1}u)(s,x)=0, u(0,\cdot)=u_{0}(\cdot)\inH^{\mu0}(\mathb {R}^{n}) \end{ar y}(3.2)

for (s, x)\in[0, S)\cross \mathbb{R}^{n} , where H^{\mu 0}(\mathbb{R}^{n}) denotes the Sobolev space of order \mu_{0}\geq 0.
Since u=u(t, \cdot) is a global solution of (3.1) if it exists on [0, T_{0} ), we say u=u(s, \cdot)=u(t(s), \cdot) is a global solution of (3.2) if it exists on [0, S_{0} ).
Let us consider the well‐posedness of (3.2). For any real numbers 2\leq q\leq\infty and2\leq r<\infty , we say that the pair (q, r) is admissible if it satisfies 1/r+2/nq=1/2.
For \mu_{0}\geq 0 and two admissible pairs \{(q_{j}, r_{j})\}_{j=1,2} , we define a function space

X^{\dot{\mu}_{0} ([0, S)):= \{u\in C([0, S), H^{\mu 0}(\mathbb{R}^{n}) ;\mu 0, \mu 0\max_{=}\Vert u\Vert_{X^{\mu}([0,S) }<\infty\}
with a metric d(u, v) :=\Vert u-v\Vert_{X^{0}([0,S))} for u, v\in X^{\mu 0}([0, S)) , where\Vertu\Vert_{X([0,S)}\mu:=\{begin{ar y}{l \Vert_{U}\Vert_{L^\infty}(0,S)L^{2}(\mathb{R}^n)\cap\bigcap_{j=1},{_2} L^{q_j}(0,S)L^{r_j}(\mathb{R}^n)} if\mu=0, \Vert_{U}\Vert_{L^\infty}(0,S)\dot{H}^\mu}(\mathb{R}^n)\cap\bigcap_{j=1} ,{_2}L^{q_j}(0,S)\dot{B}_r{j}2^{\mu}(\mathb{R}^n)} if\mu>0. \end{ar y}Here, \dot{H}^{\mu}(\mathbb{R}^{n}) and \dot{B}_{r_{j}2}^{\mu}(\mathb {R}^{n}) denote the homogeneous Sobolev and Besov spaces,

respectively. Since the propagator of the linear part of the first equation in (3.2) iswritten as \exp(\pm i\hslash\exp(-2i\omega)s\triangle/2m) , we assume 0\leq\pm\omega\leq\pi/2 to define it as a

pseudo‐differential operator. We note that the scaling critical number of p for (3.2)
is p(\mu_{0}) :=1+4/(n-2\mu_{0}) when a(\cdot)=1 . We putp_{1}( \mu_{0}):=1+\frac{4}{n-2\mu_{0} \cdot(1+\frac{4}{n-2\mu_{0} \cdot\frac{2 \mu_{0} {n(1+\sigma)})^{-1}for\sigma\neq-1.
We have the following results for time‐local, time‐global and blowing‐up solutions
for the problem (3.2).

Theorem 3.1. Let n\geq 1 , \lambda\in \mathbb{C}, 0\leq\mu_{0}<n/2 , and 1\leq p\leq p(\mu_{0}) . Let \omega satisfies
0\leq\pm\omega\leq\pi/2 and \omega\neq-\pi/2 . Assume \mu_{0}<p if p is not an odd number. Thereexist two admissible pairs \{(q_{j}, r_{j})\}_{j=1,2} with the following properties.

34(1) (Local solutions.) For any u_{0}\in H^{\mu_{0}}(\mathbb{R}^{n}) , there exist S>0 with S\leq S_{0} anda unique local solution u of (3.2) in X^{\mu 0}([0, S)) . Here, S depends only on the norm
\Vert u_{0}\Vert_{\dot{H}^{\mu_{0} (\mathb {R}^{n})} when p<p(\mu_{0}) , while S depends on the profile of u_{0} when p=p(\mu_{0}) .
The solutions depend on the initial data continuously.
(2) (Small global solutions.) Assume that one of the following conditions from(i) to (vi) holds: (i) \mu_{0}=0, p=p(0) , (ii) \mu_{0}>0, p=p(l^{x_{0}}), a_{1}\geq 0 , (iii)
1<p<p(\mu_{0}), a_{1}>0, \sigma<-1 , (iv) 1<p<p_{1}(\mu_{0}), a_{1}<0, \sigma>-1 , (v)p_{1}(\mu_{0})<p<p(\mu_{0}), a_{1}>0, \sigma>-1 , (vi) \mu_{0}>0,1<p<p(\mu_{0}), a_{1}>0, \sigma=-1.
If \Vert u_{0}\Vert_{\dot{H}^{\mu}0(\mathb {R}^{n})} is sufficiently small, then the solution u obtained in (1) is a global
solution, namely, S=S_{0}.

Corollary 3.2. Let \mu_{0}=0 or \mu_{0}=1 . Let \lambda>0 . Let 1\leq p<1+4/n when\mu_{0}=0 . Let 1\leq p<1+4/(n-2) and a_{1}(p-1-4/n)\geq 0 when \mu_{0}=1 . For anyu_{0}\in H^{\mu 0}(\mathbb{R}^{n}) , the local solution u given by (1) in Theorem 3.1 is a global solution.

Corollary 3.3. Let \mu_{0}=1, \lambda<0, a_{1}\geq 0 and 1\leq p<1+4/n . Let \omega=0 or\omega=\pi/2 . For any u_{0}\in H^{1}(\mathbb{R}^{n}) , the local solution u given by (1) in Theorem 3.1 isa global solution.
Corollary 3.4. Let \mu_{0}=1 and \lambda<0 . Let \omega\neq 0, \pi/2 . Put p_{0} :=2/(\sin 2\omega)^{2}-1.
Let p_{0}<p\leq 1+4/(n-2) . Let a_{1}(p-1-4/n)\leq 0 and S_{0}=\infty . For anyu_{0}\in H^{{\imath} (\mathbb{R}^{n}) with negative energy\int_{\mathb {R}^{n} \frac{1}{2}|\nabla u_{0}(x)|^{2}+\frac{\lambda a_{0}^{2} |u_{0}(x)|^{p+1} {p+1}dx<0 ,(3.3)

the solution u given by (1) in Theorem 3.1 blows up in finite time.
Corollary 3.5. Let \mu_{0}=1 and \lambda<0 . Let \omega=0 or \omega=\pi/2 . Let

1+4/n\leq
p\leq 1+4/(n-2) . Let a_{1}\leq 0 and S_{0}=\infty . For any u_{0}\in H^{1}(\mathbb{R}^{n}) which satisfies\Vert|x|u_{0}(x)\Vert_{L_{x}^{2}(\mathbb{R}^{n})}<\infty and (3.3), the solution u given by (1) in Theorem 3.1 blows

up in finite time.

4

Equations for vectors

So far, we have considered the partial differential equations for scalars. Our equa‐tions (1.8) and (1.9) are obtained as the Euler‐Lagrange equation for a Lagrangianand its non‐relativistic limit. To consider the equations for vectors such as Navier‐Stokes equations and elastic wave equations, we are based on the classical method

by Landau and Eckart. We note that the stress‐energy tensor T_{\beta}^{\alpha} must satisfy theconservation law\nabla_{\alpha}T_{\beta}^{\alpha}=0
in the Einstein equation (2.4). We introduce the stress tensor P^{\alpha\beta} . Let \lambda, \mu be twoconstants. Let p be the pressure. Let v^{\alpha} be a contravariant tensor which satisfies

35\lim_{carrow\infty}\partial_{j}v^{0}=0 . Put

P^{\alpha\beta}:=-pg^{\alpha\beta}+\lambda g^{\alpha\beta}\nabla_{\gamma} v^{\gamma}+\mu(\nabla^{\alpha}v^{\beta}+\nabla^{\beta}v^{\alpha}) .

(4.1)

Then the nonrelativistic limit yields the equation\lim_{car ow\infty}\nabla_{\alpha}P^{\alpha\beta}=-\partial^{\beta}p+ \mu\partial_{j}\partial^{j}v^{\beta}+(\mu+\lambda)\partial^{\beta}\partial_{j}v^ {j},where we regard \partial^{0} :=-1\partial=0 in the RHS.

4.1

Navier‐Stokes equations

Let us consider the Navier‐Stokes equation. Since any velocity tensor u^{\alpha} must satisfy-c^{2}=-c^{2}(u^{0})^{2}+a(x^{0})^{2} \sum_{j=1}^{n}(u^{j})^{2},
we have \lim_{carrow\infty}u^{0}=\pm 1 . Based on this, we assume

car ow\infty 1\dot{ \imath} mu^{0}=1 and cl ım\infty\partial\alpha u0 =0 for 0\leq\alpha\leq n .
arrow

(4.2)

Let P^{\alpha\beta} be the stress tensor with v^{\alpha} :=u^{\alpha} . Put

T^{\alpha\beta}:=( \rho+\frac{p}{c^{2} )u^{\alpha}u^{\beta}-P^{\alpha\beta} .

(4.3)

Put f^{k} :=u^{j}\partial_{j}u^{k} . Let \rho(x^{0}) :=C/a(x^{0})^{n} (the density of mass). Then\lim_{car ow\infty}\nabla_{\alpha}T^{\alpha\beta}=0(4.4)\partial_{j}u^{j}=0(4.5)\partial_{0}u^{k}+f^{k}+\frac{\partial_{0}a^{2} {a^{2} u^{k}+\frac{1}{\rho} \partial^{k}p-\frac{\mu}{\rho}\partial_{j}\partial^{j}u^{k}=0 .

(4.6)\partial_{0}e^{0}+\partial_{j}e^{j}+e_{*}=0(4.7)

are equivalent toandFor the equation (4.6) with (4.5), the energy estimate

holds, wheree^{0}:= \frac{1}{2}u_{k}u^{k}, J^{j}:=\frac{1}{2}u^{j}u_{k}u^{k}-u^{j}\partial_ {k}\frac{1}{\partial^{\el }\partial_{\el } f^{k}+\partial_{k}u^{k}\frac{1} {\partial^{\el }\partial_{\el } f^{j},e^{j}:=J^{j}- \frac{\mu}{\rho}u_{k}\partial^{g}u^{k}, K:=\frac{1}{2} (\frac{\partial_{0}a^{2} {a^{2} -\partial_{j}u^{j})Here,

K satisfies,

e_{*} :=Ku_{k}u^{k}+ \frac{\mu}{p}\partial^{\el }u_{k}\partial_{\el }u^{k}

K= \frac{\partial_{0}a^{2} {2a^{2} .

(4.8)

36
4.2

The elastic wave equations

Let us consider the elastic wave equations. For the displacement tensor r^{\alpha} , and its
relativistic velocity u^{\alpha} :=dr^{\alpha}/d\tau , we put

dr^{\alpha}:=r^{\alpha}(\tau+d\tau)-r^{\alpha}(\tau).

Since dr must satisfy-c^{2}(d \tau)^{2}=-c^{2}(dr^{0})^{2}+a(x^{0})^{2}\sum_{j=1}^{n}(drj)2,

we have \lim_{carrow\infty}dr^{0}/d\tau=\pm 1 . Based on this, we assume\lim_{carrow\infty}r^{0}=x^{0} , and cl ım\infty\partialjr0 =0 for 1\leq j\leq n .
arrow

(4.9)

Let P^{\alpha\beta} be the stress tensor with v^{\alpha} :=r^{\alpha} . We put u^{\alpha} :=dr^{\alpha}/d\tau,T^{\alpha\beta} :=( \rho+\frac{p}{c^{2} )u^{\alpha}u^{\beta}-P^{\alpha\beta} ,(4.10)and h^{k} :=\partial_{0}r^{j}\partial_{j}\partial_{0}r^{k} . Put \rho(x^{0}) :=C/a(x^{0})^{n} (density of mass). Then\lim_{car ow\infty}\nabla_{\alpha}T^{\alpha\beta}=0(4.11)\partial_{0}\partial_{j}r^{j}=0(4.12)

are equivalent toand\partial_{0}^{2}r^{k}+h^{k}+\frac{\partial_{0}a^{2} {a^{2} \partial_{0}r^{k}+ \frac{1}{\rho}\partial^{k}p-\frac{\mu}{\rho}\partial_{j}\partial^{g}r^{k}-\frac{ \mu+\lambda}{\rho}\partial^{k}\partial_{j}r^{j}=0 .

(4.13)

We put(\partial_{\tau}r)_{\alpha}:=g_{\alpha\beta}\partial_{\tau}r^{\beta}, K:= \underline{\parti2a^{2}al_{0}a^{2}.

(4.14)For the equation (4.13) with (4.12), the energy estimate\partial_{0}e^{0}+\partial_{j}e^{j}+e_{*}=0(4.15)

holds, wheree^{0}:= \frac{1}{2}(\partial_{0}r)_{k}\partial_{0}r^{k}+\frac{\mu}{2\rho} \partial_{j}r_{k}\partial^{g}r^{k}-\frac{\mu}{2\rho}(\partial_{j}r^{j})^{2},e^{j}:= \frac{1}{2}(\partial_{0}r^{j})(\partial_{0}r)_{k}\partial_{0}r^{k}- \partial_{0}r^{j}\partial_{k}\frac{1}{\partial_{\el }\partial^{\el } h^{k}+ \partial^{k}(\partial_{0}r)_{k}\frac{1}{\partial_{\el }\partial^{\el } h^{j}- \frac{\mu}{\rho}(\partial_{0}r)_{k}\partial^{g}r^{k}+\frac{I^{I} {\rho} \partial_{0}r^{J}\acute{\partial}_{k}r^{k},e_{*} :=K \cdot(\partial_{0}r)_{k}\partial_{0}r^{k}+\frac{4\mu}{\rho}\cdot K\partial_{\el }r_{k}\partial^{\el }r^{k}.

37
Remark 4.1. In the energy estimates (4.7) and (4.15) , the energy density e_{*} shows
the dissipative or anti‐dissipative property. When the space is expanding, namely,\partial_{0}a(\cdot)>0 , then K is a positive function, which shows that e_{*} is positive for non‐zero velocity u and non‐constant displacement r . From this fact, we expect the strong
dissipative effects on the solutions of the equations by the spatial expansion.
Acknowledgment. The author is thankful to Professor Yoshio Tsutsumi for
the invitation to the conference “RIMS Workshop: Nonlinear Wave and Dispersive
Equations

References[1] Y. Choquet‐Bruhat, Results and open problems in mathematical general relativity, Mi‐
lan J. Math. 75 (2007), 273‐289.
[2] Y. Choquet‐Bruhat, General relativity and the Einstein equations, Oxford Mathemati‐
cal Monographs, Oxford University Press, Oxford, 2009, xxvi+785 pp.

[3] H. \Gamma . M. Goenner, On the History of Unified Field Theor es, Living Rev. Relativity 7(2004), 2.

[4] H. F. M. Goenner, On the History of Unified Field Theones. Part II. (ca. 1930−ca.
1965), Living Rev. Relativity 17 (20ı4), 5.
[5] T. Kaluza, Zum Unitätsproblem in der Physik, Sitzungsber Preuss. Akad. Wiss. Berlin.
(Math. Phys.) (1921), 966‐972.

[6] O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Zeitschrift für
Physik A. 37 (12) (ı926), 895‐906.