Heat equation and smoothness effect

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Consider the homogeneous heat system with Dirichlet B.C



beginarrayc
u_t-u_xx=0 \
u(0,t)=u(pi ,t)=0 \
u(x,0)=f(x).%
endarray
It is well known that the solution of the system above is given by



$$u(t,x)=sum_ngeq 1A_ne^-n^2tsin (nx)$$,



Also, it is well known that $uin C^infty ((0,T]times (0,pi ))$ for any
$fin L^2(0,pi ).$ (Smoothness effect).



My question is : What if there exists some function $fin X$ such that $%
A_nsim e^n^2,$ where $X$ is some Hilbert or Banach space, the
solution will not exist,



Is there any initial condition and some $X$ which satisfy that ?



Thanks.




pde regularity-theory-of-pdes parabolic-pde




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    up vote
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    down vote

    favorite
    2












    Consider the homogeneous heat system with Dirichlet B.C



    beginarrayc
    u_t-u_xx=0 \
    u(0,t)=u(pi ,t)=0 \
    u(x,0)=f(x).%
    endarray
    It is well known that the solution of the system above is given by



    $$u(t,x)=sum_ngeq 1A_ne^-n^2tsin (nx)$$,



    Also, it is well known that $uin C^infty ((0,T]times (0,pi ))$ for any
    $fin L^2(0,pi ).$ (Smoothness effect).



    My question is : What if there exists some function $fin X$ such that $%
    A_nsim e^n^2,$ where $X$ is some Hilbert or Banach space, the
    solution will not exist,



    Is there any initial condition and some $X$ which satisfy that ?



    Thanks.




    pde regularity-theory-of-pdes parabolic-pde




    share|cite|improve this question





















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      Consider the homogeneous heat system with Dirichlet B.C



      beginarrayc
      u_t-u_xx=0 \
      u(0,t)=u(pi ,t)=0 \
      u(x,0)=f(x).%
      endarray
      It is well known that the solution of the system above is given by



      $$u(t,x)=sum_ngeq 1A_ne^-n^2tsin (nx)$$,



      Also, it is well known that $uin C^infty ((0,T]times (0,pi ))$ for any
      $fin L^2(0,pi ).$ (Smoothness effect).



      My question is : What if there exists some function $fin X$ such that $%
      A_nsim e^n^2,$ where $X$ is some Hilbert or Banach space, the
      solution will not exist,



      Is there any initial condition and some $X$ which satisfy that ?



      Thanks.




      pde regularity-theory-of-pdes parabolic-pde




      share|cite|improve this question











      Consider the homogeneous heat system with Dirichlet B.C



      beginarrayc
      u_t-u_xx=0 \
      u(0,t)=u(pi ,t)=0 \
      u(x,0)=f(x).%
      endarray
      It is well known that the solution of the system above is given by



      $$u(t,x)=sum_ngeq 1A_ne^-n^2tsin (nx)$$,



      Also, it is well known that $uin C^infty ((0,T]times (0,pi ))$ for any
      $fin L^2(0,pi ).$ (Smoothness effect).



      My question is : What if there exists some function $fin X$ such that $%
      A_nsim e^n^2,$ where $X$ is some Hilbert or Banach space, the
      solution will not exist,



      Is there any initial condition and some $X$ which satisfy that ?



      Thanks.





      pde regularity-theory-of-pdes parabolic-pde





      share|cite|improve this question










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      asked Jul 31 at 16:36









      Gustave

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