Definition edit

Let ${\displaystyle A}$  and ${\displaystyle B}$  be two linear operators, with domains ${\displaystyle D(A)}$  and ${\displaystyle D(B)}$ , acting in a Banach space ${\displaystyle X}$ .^[4][5][6] A function ${\displaystyle u(t):[0,T]\to X}$  is said to have strong derivative (or to be Frechet differentiable or simply differentiable) at the point ${\displaystyle t_{0}}$  if there exists an element ${\displaystyle y\in X}$  such that

${\displaystyle \lim _{h\to 0}\left\|{\frac {u(t_{0}+h)-u(t_{0})}{h}}-y\right\|=0}$ 

and its derivative is ${\displaystyle u'(t_{0})=y}$ .

A solution of the equation

${\displaystyle B{\frac {\mathrm {d} u}{\mathrm {d} t}}=Au}$ 

is a function ${\displaystyle u(t):[0,\infty )\to D(A)\cap D(B)}$  such that:

• ${\displaystyle (Bu)(t)\in C([0,\infty );X),}$ 
• the strong derivative ${\displaystyle u'(t)}$  exists ${\displaystyle \forall t\in [0,\infty )}$  and ${\displaystyle u'(t)\in D(B)}$  for any such ${\displaystyle t}$ , and
• the previous equality holds ${\displaystyle \forall t\in [0,\infty )}$ .

The Cauchy problem consists in finding a solution of the equation, satisfying the initial condition ${\displaystyle u(0)=u_{0}\in D(A)\cap D(B)}$ .

Well posedness edit

According to the definition of well-posed problem by Hadamard, the Cauchy problem is said to be well posed (or correct) on ${\displaystyle [0,\infty )}$  if:

• for any ${\displaystyle u_{0}\in D(A)\cap D(B)}$  it has a unique solution, and
• this solution depends continuously on the initial data in the sense that if ${\displaystyle u_{n}(0)\to 0}$  (${\displaystyle u_{n}(0)\in D(A)\cap D(B)}$ ), then ${\displaystyle u_{n}(t)\to 0}$  for the corresponding solution at every ${\displaystyle t\in [0,\infty ).}$ 

A well posed Cauchy problem is said to be uniformly well posed if ${\displaystyle u_{n}(0)\to 0}$  implies ${\displaystyle u_{n}(t)\to 0}$  uniformly in ${\displaystyle t}$  on each finite interval ${\displaystyle [0,T]}$ .

Semigroup of operators associated to a Cauchy problem edit

To an abstract Cauchy problem one can associate a semigroup of operators ${\displaystyle U(t)}$ , i.e. a family of bounded linear operators depending on a parameter ${\displaystyle t}$  (${\displaystyle 0<t<\infty }$ ) such that

${\displaystyle U(t_{1}+t_{2})=U(t_{1})U(t_{2})\quad (0<t_{1},t_{2}<\infty ).}$ 

Consider the operator ${\displaystyle U(t)}$  which assigns to the element ${\displaystyle u_{n}(0)\in D(A)\cap D(B)}$  the value of the solution ${\displaystyle u(t)}$  of the Cauchy problem (${\displaystyle u(0)=u_{0}}$ ) at the moment of time ${\displaystyle t>0}$ . If the Cauchy problem is well posed, then the operator ${\displaystyle U(t)}$  is defined on ${\displaystyle D(A)\cap D(B)}$  and forms a semigroup.

Additionally, if ${\displaystyle D(A)\cap D(B)}$  is dense in ${\displaystyle X}$ , the operator ${\displaystyle U(t)}$  can be extended to a bounded linear operator defined on the entire space ${\displaystyle X}$ . In this case one can associate to any ${\displaystyle x_{0}\in X}$  the function ${\displaystyle U(t)x_{0}}$ , for any ${\displaystyle t>0}$ . Such a function is called generalized solution of the Cauchy problem.

If ${\displaystyle D(A)\cap D(B)}$  is dense in ${\displaystyle X}$  and the Cauchy problem is uniformly well posed, then the associated semigroup ${\displaystyle U(t)}$  is a C₀-semigroup in ${\displaystyle X}$ .

Conversely, if ${\displaystyle A}$  is the infinitesimal generator of a C₀-semigroup ${\displaystyle U(t)}$ , then the Cauchy problem

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au\quad u(0)=u_{0}\in D(A)}$ 

is uniformly well posed and the solution is given by

${\displaystyle u(t)=U(t)u_{0}.}$ 

The Cauchy problem

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au+f\quad u(0)=u_{0}\in D(A)}$ 

with ${\displaystyle f:[0,\infty )\to X}$ , is called nonhomogeneous when ${\displaystyle f(t)\neq 0}$ . The following theorem gives some sufficient conditions for the existence of the solution:

Theorem. If ${\displaystyle A}$  is an infinitesimal generator of a C₀-semigroup ${\displaystyle T(t)}$  and ${\displaystyle f}$  is continuously differentiable, then the function

${\displaystyle u(t)=T(t)u_{0}+\int _{0}^{t}T(t-s)f(s)\,ds,\quad t\geq 0}$ 

is the unique solution to the (abstract) nonhomogeneous Cauchy problem.

The integral on the right-hand side as to be intended as a Bochner integral.

The problem^[7] of finding a solution to the initial value problem

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=A(t)u+f\quad u(0)=u_{0}\in D(A),}$ 

where the unknown is a function ${\displaystyle u:[0,T]\to X}$ , ${\displaystyle f:[0,T]\to X}$  is given and, for each ${\displaystyle t\in [0,T]}$ , ${\displaystyle A(t)}$  is a given, closed, linear operator in ${\displaystyle X}$  with domain ${\displaystyle D[A(t)]=D}$ , independent of ${\displaystyle t}$  and dense in ${\displaystyle X}$ , is called time-dependent Cauchy problem.

An operator valued function ${\displaystyle U(t,\tau )}$  with values in ${\displaystyle B(X)}$  (the space of all bounded linear operators from ${\displaystyle X}$  to ${\displaystyle X}$ ), defined and strongly continuous jointly in ${\displaystyle t,\tau }$  for ${\displaystyle 0\leq \tau \leq t\leq T}$ , is called a fundamental solution of the time-dependent problem if:

• the partial derivative ${\displaystyle {\frac {\mathrm {\delta } U(t,\tau )}{\mathrm {\delta } t}}}$  exists in the strong topology of ${\displaystyle X}$ , belongs to ${\displaystyle B(X)}$  for ${\displaystyle 0\leq \tau \leq t\leq T}$ , and is strongly continuous in ${\displaystyle t}$  for ${\displaystyle 0\leq \tau \leq t\leq T}$ ;
• the range of ${\displaystyle U(t,\tau )}$  is in ${\displaystyle D}$ ;
• ${\displaystyle {\frac {\mathrm {\delta } U(t,\tau )}{\mathrm {\delta } t}}+A(t)U(t,\tau )=0,\quad 0\leq \tau \leq t\leq T,}$  and
• ${\displaystyle U(\tau ,\tau )=I}$ .

${\displaystyle U(\tau ,\tau )}$  is also called evolution operator, propagator, solution operator or Green's function.

A function ${\displaystyle u:[0,T]\to X}$  is called a mild solution of the time-dependent problem if it admits the integral representation

${\displaystyle u(t)=U(t,0)u_{0}+\int _{0}^{t}U(t,s)f(s)\,ds,\quad t\geq 0.}$ 

There are various known sufficient conditions for the existence of the evolution operator ${\displaystyle U(t,\tau )}$ . In practically all cases considered in the literature ${\displaystyle -A(t)}$  is assumed to be the infinitesimal generator of a C₀-semigroup on ${\displaystyle X}$ . Roughly speaking, if ${\displaystyle -A(t)}$  is the infinitesimal generator of a contraction semigroup the equation is said to be of hyperbolic type; if ${\displaystyle -A(t)}$  is the infinitesimal generator of an analytic semigroup the equation is said to be of parabolic type.

The problem^[7] of finding a solution to either

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=f(t,u)\quad u(0)=u_{0}\in X}$ 

where ${\displaystyle f:[0,T]\times X\to X}$  is given, or

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=A(t)u\quad u(0)=u_{0}\in D(A)}$ 

where ${\displaystyle A}$  is a nonlinear operator with domain ${\displaystyle D(A)\in X}$ , is called nonlinear Cauchy problem.

• Cauchy problem
• C0-semigroup