Calculation of the integral

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Let $M$ is a noncompact manifold.



Let $R=overlineB(x_0;rho)times [-tau_1,-tau_2]$. ($B(x_0;rho)$ is the ball of the radius $rho$ centered at $x_0$).



And $w$ satisfies $wgeq0, Lwgeq0$ ($L$ is the heat operator on $M$).
Let $psi$ is a function.
Assume that the following holds:



$frac12int!!!int_R psi^2 fracpartial w^2partial t+mid grad(w) mid ^2 dVdt leq 2 int !!!int_R mid grad(psi )mid^2w^2dVdt$ $cdots$(1).



Then



$int!!!int_R fracpartial (psi^2 w^2)partial t + mid grad(psi w) mid^2 dVdt leq c int!!!int_R psi fracpartial psipartial t + mid grad(psi) mid^2 w^2 dVdt$ $cdots$(2),



where c is some universal constant.



Why does the inequality (1) imply the inequality (2)?



I would appreciate it if you could answer this question.




integration inequality heat-equation




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    up vote
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    down vote

    favorite












    Let $M$ is a noncompact manifold.



    Let $R=overlineB(x_0;rho)times [-tau_1,-tau_2]$. ($B(x_0;rho)$ is the ball of the radius $rho$ centered at $x_0$).



    And $w$ satisfies $wgeq0, Lwgeq0$ ($L$ is the heat operator on $M$).
    Let $psi$ is a function.
    Assume that the following holds:



    $frac12int!!!int_R psi^2 fracpartial w^2partial t+mid grad(w) mid ^2 dVdt leq 2 int !!!int_R mid grad(psi )mid^2w^2dVdt$ $cdots$(1).



    Then



    $int!!!int_R fracpartial (psi^2 w^2)partial t + mid grad(psi w) mid^2 dVdt leq c int!!!int_R psi fracpartial psipartial t + mid grad(psi) mid^2 w^2 dVdt$ $cdots$(2),



    where c is some universal constant.



    Why does the inequality (1) imply the inequality (2)?



    I would appreciate it if you could answer this question.




    integration inequality heat-equation




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $M$ is a noncompact manifold.



      Let $R=overlineB(x_0;rho)times [-tau_1,-tau_2]$. ($B(x_0;rho)$ is the ball of the radius $rho$ centered at $x_0$).



      And $w$ satisfies $wgeq0, Lwgeq0$ ($L$ is the heat operator on $M$).
      Let $psi$ is a function.
      Assume that the following holds:



      $frac12int!!!int_R psi^2 fracpartial w^2partial t+mid grad(w) mid ^2 dVdt leq 2 int !!!int_R mid grad(psi )mid^2w^2dVdt$ $cdots$(1).



      Then



      $int!!!int_R fracpartial (psi^2 w^2)partial t + mid grad(psi w) mid^2 dVdt leq c int!!!int_R psi fracpartial psipartial t + mid grad(psi) mid^2 w^2 dVdt$ $cdots$(2),



      where c is some universal constant.



      Why does the inequality (1) imply the inequality (2)?



      I would appreciate it if you could answer this question.




      integration inequality heat-equation




      share|cite|improve this question











      Let $M$ is a noncompact manifold.



      Let $R=overlineB(x_0;rho)times [-tau_1,-tau_2]$. ($B(x_0;rho)$ is the ball of the radius $rho$ centered at $x_0$).



      And $w$ satisfies $wgeq0, Lwgeq0$ ($L$ is the heat operator on $M$).
      Let $psi$ is a function.
      Assume that the following holds:



      $frac12int!!!int_R psi^2 fracpartial w^2partial t+mid grad(w) mid ^2 dVdt leq 2 int !!!int_R mid grad(psi )mid^2w^2dVdt$ $cdots$(1).



      Then



      $int!!!int_R fracpartial (psi^2 w^2)partial t + mid grad(psi w) mid^2 dVdt leq c int!!!int_R psi fracpartial psipartial t + mid grad(psi) mid^2 w^2 dVdt$ $cdots$(2),



      where c is some universal constant.



      Why does the inequality (1) imply the inequality (2)?



      I would appreciate it if you could answer this question.





      integration inequality heat-equation





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 21 at 3:06









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